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14 More on Functions

Topics Covered

In case you missed something in class, or just want to review a specific topic covered in this Module, here is a list of topics covered:
  1. Identify Basic (Toolkit) Functions
  2. Determining Even and Odd Functions
  3. Difference Quotient and Slope
  4. Key Concepts

Identify Graphs of Basic (Toolkit) Functions

We can write y = 2x − 3 as in function notation as f(x) = 2x − 3. It still means the same thing. The graph of the function is the graph of all ordered pairs (x, y) where y = f(x). So we can write the ordered pairs as (x, f(x)). It looks different but the graph will be the same.

This figure has a graph next to a table. The graph has a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled f of x equals2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equalsnegative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label f of x equals2 x minus 3. The second row is a header row with the headers x, f of x, and (x, f of x). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5).

Graph of a Function

The graph of a function is the graph of all its ordered pairs, (x, y) or using function notation, (x, f(x)) where y = f(x).

Notation description
f name of function
x x-coordinate of the ordered pair
f(x) y-coordinate of the ordered pair

 

As we move forward in our study, it is helpful to be familiar with the graphs of several basic functions and be able to identify them.

We wrote linear equations in several forms, but it will be most helpful for us here to use the slope-intercept form of the linear equation. The slope-intercept form of a linear equation is y = mx + b. In function notation, this linear function becomes f(x) = mx + b where m is the slope of the line and b is the y-intercept. The domain is the set of all real numbers, and the range is also the set of all real numbers.

Linear Function

This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equals m x plus b”, “m, b: all real numbers”, “m: slope of the line”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.

We will use the graphing techniques we used earlier, to graph the basic functions.

Try it!

Graph: f(x) = −2x − 4.

Solution:

Steps Algebraic 
f(x) = −2x − 4
We recognize this as a linear function.
Find the slope and y-intercept. m = −2, b = −4
Graph using the slope intercept. Graph of line passing through (-2,0) and (0,-4)

The next function whose graph we will look at is called the constant function and its equation is of the form f(x) = b, where b is any real number. If we replace the f(x) with y, we get y = b. We recognize this as the horizontal line whose y-intercept is b. The graph of the function f(x) = b, is also the horizontal line whose y-intercept is b.

Notice that for any real number we put in the function, the function value will be b. This tells us the range has only one value, b.

Constant Function

This figure has a graph of a straight horizontal line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equals b”, “b: any real number”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: b”.

Try it!

Graph: f(x) = 4.

Solution:

Steps Algebraic 
f(x) = 4
We recognize this as a constant function.
The graph will be a horizontal line through

(0, 4).

 The graph will be a horizontal line through (0,4).

Graph: f(x) = −2.

Solution:

The figure has the graph of a constant function on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (1, negative 2), and (2, negative 2).

The identity function, f(x) = x is a special case of the linear function. If we write it in linear function form, f(x) = 1x + 0, we see the slope is 1 and the y-intercept is 0.

Identity Function

This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the points (0, 0), (1, 1), and (2, 2). Next to the graph are the following: “f of x equalsx”, “m: 1”, “b: 0”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.

The next function we will look at is not a linear function. So the graph will not be a line. The only method we have to graph this function is point plotting. Because this is an unfamiliar function, we make sure to choose several positive and negative values as well as 0 for our x-values.

Try it! – Non-Linear Function

Graph: f(x) = x2.

Solution:

We choose x-values. We substitute them in and then create a chart as shown.

This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9). The table has 8 rows and 3 columns. The first row is a header row with the headers x, f of x equalsx squared, and (x, f of x). The second row has the coordinates negative 3, 9, and (negative 3, 9). The third row has the coordinates negative 2, 4, and (negative 2, 4). The fourth row has the coordinates negative 1, 1, and (negative 1, 1). The fifth row has the coordinates 0, 0, and (0, 0). The sixth row has the coordinates 1, 1, and (1, 1). The seventh row has the coordinates 2, 4, and (2, 4). The seventh row has the coordinates 3, 9, and (3, 9).

Looking at the result above, we can summarize the features of the square (Quadratic) function. We call this graph a parabola. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.

The range is not all real numbers. Notice the graph consists of values of y never go below zero. This makes sense as the square of any number cannot be negative. So, the range of the square function is all non-negative real numbers.

Square Function

This figure has a graph of a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4). Next to the graph are the following: “f of x equalsx squared”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.

The next function we will look at is also not a linear function so the graph will not be a line. Again we will use point plotting, and make sure to choose several positive and negative values as well as 0 for our x-values.

Try it! -Non-linear Function

Graph: f(x) = x3.

Solution:

We choose x-values. We substitute them in and then create a chart.

This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8). Next to the graph is a table. The table has 6 rows and 3 columns. The first row is a header row with the headers x, f of x equalsx cubed, and (x, f of x). The second row has the coordinates negative 2, negative 8, and (negative 2, negative 8). The third row has the coordinates negative 1, negative 1, and (negative 1, negative 1). The fourth row has the coordinates 0, 0, and (0, 0). The fifth row has the coordinates 1, 1, and (1, 1). The sixth row has the coordinates 2, 8, and (2, 8).

2. Graph: f(x) = −x3.

Solution:

This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, 8), (negative 1, 1), (0, 0), (1, negative 1), and (2, negative 8).

Looking at the result in the first example on cube function above, we can summarize the features of the cube function. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.

The range is all real numbers. This makes sense as the cube of any non-zero number can be positive or negative. So, the range of the cube function is all real numbers.

Cube Function

This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).). Next to the graph are the following: “f of x equalsx cubed”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.

The next function we will look at does not square or cube the input values, but rather takes the square root of those values.

Let’s graph the function \displaystyle f(x) = \sqrt{x} and then summarize the features of the function. Remember, we can only take the square root of non-negative real numbers, so our domain will be the non-negative real numbers.

Try it!- Square root function

\displaystyle f(x) = \sqrt{x}

Solution:

We choose x-values. Since we will be taking the square root, we choose numbers that are perfect squares, to make our work easier. We substitute them in and then create a chart.

This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph is a table. The table has 5 rows and 3 columns. The first row is a header row with the headers x, f of x equalssquare root of x, and (x, f of x). The second row has the coordinates 0, 0, and (0, 0). The third row has the coordinates 1, 1, and (1, 1). The fourth row has the coordinates 4, 2, and (4, 2). The fifth row has the coordinates 9, 3, and (9, 3).

2. Graph: \displaystyle f(x) = -\sqrt{x}

Solution:

This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from negative 10 to 0. The curved half-line starts at the point (0, 0) and then goes down and to the right. The curved half line goes through the points (1, negative 1), (4, negative 2), and (9, negative 3).

Square Root Function

[0, infinity)”, and “Range: [0, infinity)”.” width=”404″ height=”252″ />

Our last basic function is the absolute value function, f(x) = |x|. Keep in mind that the absolute value of a number is its distance from zero. Since we never measure distance as a negative number, we will never get a negative number in the range.

Try It!- Absolute Value Function

Graph: f(x) = |x|.

Solution:

We choose x-values. We substitute them in and then create a chart.

This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). Next to the graph is a table. The table has 8 rows and 3 columns. The first row is a header row with the headers x, f of x equalsabsolute value of x, and (x, f of x). The second row has the coordinates negative 3, 3, and (negative 3, 3). The third row has the coordinates negative 2, 2, and (negative 2, 2). The fourth row has the coordinates negative 1, 1, and (negative 1, 1). The fifth row has the coordinates 0, 0, and (0, 0). The sixth row has the coordinates 1, 1, and (1, 1). The seventh row has the coordinates 2, 2, and (2, 2). The eighth row has the coordinates 3, 3, and (3, 3).

Graph: f(x) = −|x|.

Solution:

This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 8 to 4. The v-shaped line goes through the points (negative 3, negative 3), (negative 2, negative 2), (negative 1, negative 1), (0, 0), (1, negative 1), (2, negative 2), and (3, negative 3).

Absolute Value Function

This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). The point (0, 0) where the line changes slope is called the vertex. Next to the graph are the following: “f of x equalsabsolute value of x”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.

Determining Even and Odd Functions

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions f(x) = x2  or f(x) = | x | will result in the original graph. We say that these types of graphs are symmetric about the y-axis. A function whose graph is symmetric about the y-axis is called an even function.

If the graphs of f(x) = x3  or f(x) = \displaystyle\frac{1}{x} were reflected over both axes, the result would be the original graph, as shown below.

Graph of x^3 and its reflections.
(a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.

We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function.

Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, f(x) = 2x is neither even nor odd. Also, the only function that is both even and odd is the constant function f(x) = 0.

Even and Odd Functions

A function is called an even function if for every input x

f(x) = f(−x)

The graph of an even function is symmetric about the y-axis.

A function is called an odd function if for every input x

f(x) = −f(−x)

The graph of an odd function is symmetric about the origin.

How To: Given the formula for a function, determine if the function is even, odd, or neither.

  1. Determine whether the function satisfies f(x) = f(−x). If it does, it is even.
  2. Determine whether the function satisfies f(x) = −f(−x) If it does, it is odd.
  3. If the function does not satisfy either rule, it is neither even nor odd.

Try it! – Determining whether a Function Is Even, Odd, or Neither

Is the function f(x) = x3 + 2x even, odd, or neither?

Solution

Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let’s begin with the rule for even functions.

f(−x) = (−x)3 + 2(−x) = −x3 − 2x

This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.

−f(−x) = −(−x3 − 2x) = x3 + 2x

Because −f(−x) = f(x), this is an odd function.

Analysis Consider the graph of f below. Notice that the graph is symmetric about the origin. For every point (x, y) on the graph, the corresponding point (−x,−y) is also on the graph. For example, (1, 3) is on the graph of f, and the corresponding point (−1, −3) is also on the graph.

Graph of f(x) with labeled points at (1, 3) and (-1, -3).

Is the function a)  f(s) = s4 + 3s2 + 7,  b) h(x) = \displaystyle\frac{1}{x} + 3x even, odd, or neither?

Solution:

even

Difference Quotient

While we will look more closely at slope in the next section,

What is Slope?

The slope of a line, m, represents the vertical change in y over the horizontal change in x. Given two points, (x1, y1) and (x2, y2), the following formula determines the slope of a line containing these points:

m = \displaystyle\frac{y_{2} - y_{1}}{x_{2} - x_{1}}

Slope also indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.

We can use the slope of a line to a function at a point (a, f(a)) to estimate the rate of change, or the rate at which one variable changes in relation to another variable. We can obtain the slope by choosing a value of  x near a and drawing a line through the points (a, f(a)) and (x, f(x)). The slope of this line is given by an equation in the form of a difference quotient:

m = \displaystyle\frac{f(x) - f(a)}{x - a}

We can also calculate the slope of a line to a function at a value a by using this equation and replacing x with a + h, where h is a value close to 0. We can then calculate the slope of the line through the points (a, f(a)) and (a+h,f(a+h)). In this case, we find the line has a slope given by the following difference quotient with increment h:

m = \displaystyle\frac{f(a + h) - f(a)}{a + h - a} = \frac{f(a +h) - f(a)}{h}

Difference Quotient

Let f be a function defined on an interval I containing a. If x ≠ a is in I, then

Q = \displaystyle\frac{f(x) - f(a)}{x - a}

is a difference quotient. Also, if h ≠ 0 is chosen so that a + h is in I, then

Q = \displaystyle\frac{f(a + h) - f(a)}{h}

is a difference quotient with increment h.

Try it!

Given the function g(x) = x2 + 2x, simplify \displaystyle\frac{g(x) - g(a)}{x - a} , x ≠ a.

Solution:

\displaystyle\frac{g(x) - g(a)}{x - a} = x + a + 2, x ≠ a.

#2 Evaluating Functions at Specific Values

Evaluate f(x) = x2 + 3x − 4

  1. 2
  2. a
  3. a + h
  4. \displaystyle\frac{f(a + h) - f(a)}{h}

Solution:

Replace the x in the function with each specified value.

  1. Because the input value is a number, 2, we can use simple algebra to simplify.

    f(2) = 22 + 3(2) − 4 = 4 + 6 − 4 = 6

     

  2. In this case, the input value is a letter so we cannot simplify the answer any further.

    f(a) = a2 + 3a − 4

     

  3. With an input value of a + h, we must use the distributive property.

    f(a + h) = (a + h)2 + 3(a + h)−4 = a2 + 2ah + h2 + 3a + 3h−4

     

  4. In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that

    f(a + h) = a2 + 2ah + h2 + 3a + 3h − 4

    and we know that

    f(a) = a2 + 3a − 4

    Now we combine the results and simplify.

    \displaystyle\frac{f(a + h) - f(a)}{h} = \displaystyle\frac{(a^2 + 2ah + h^2 + 3a + 3h - 4) - (a^2 +3a - 4)}{h}

    \displaystyle\frac{f(a + h) - f(a)}{h} = \displaystyle\frac{(2ah + h^2 + 3h)}{h}

    \displaystyle\frac{f(a + h) - f(a)}{h} = \displaystyle\frac{h(2a + h + 3)}{h}

    \displaystyle\frac{f(a + h) - f(a)}{h} = 2a + h +3

     

Key Concepts

  • Even functions are symmetric about the y-axis, whereas odd functions are symmetric about the origin.
  • Even functions satisfy the condition f(x) = f(−x).
  • Odd functions satisfy the condition f(x) = −f(−x).
  • A function can be odd, even, or neither.
  • Difference quotient
    Q = \displaystyle\frac{f(x) - f(a)}{x - a}
  • Difference quotient with increment h
    Q = \displaystyle\frac{f(a + h) - f(a)}{a + h - a} = \frac{f(a +h) - f(a)}{h}
definition

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