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2. Functions & Relations
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Algebra 1

2.03 Evaluating functions

Lesson
Practice

Evaluating functions

Recall that a function maps each input of a relation to exactly one output. Functions are typically represented in function notation, so the relationship between inputs and outputs are clear.

Input

The independent variable of a function; usually the x-value

Output

The dependent variable of a function; usually the y-value

Function notation

A notation that describes a function. For a function f when x is the input, the symbol f\left(x\right) denotes the corresponding output.

We have seen the equation for a linear function y=mx+b. By naming a linear function f, you can also write the function using function notation: f\left(x\right)=mx+b.

This is a way of saying that mx+b is a function of x. This is useful because it quickly tells us that we are working with a function where y can represent a relation that is not a function.

The notation f\left(x\right) is another name for y. If f is a function, and x is in its domain, then f\left(x\right) represents the output of f corresponding to the input x. You can use letters other than f to name a function, such as g or h.

\displaystyle f\left(x\right)=y
\bm{f}
is the name of function
\bm{x}
is the input
\bm{y}
is the output

To evaluate a function at a point is to calculate the output value at a particular input value:

If f\left(x\right)=-7x+9, then determine the value of f\left(1\right).

This is the same as stating to evaluate the function y=-7x+9 when x=1.

f\left(1\right)=-7\left(1\right)+9

f\left(1\right)=-7+9=2

Therefore, f\left(1\right)=2 for the function f\left(x\right)=-7x+9.

Examples

Example 1

Consider the functionf\left(x\right) = \dfrac {x}{3}-5 where x is the independent variable.

a

Construct a table of values for the function at x=-3, \,0, \,9, \,12, \,27.

Worked Solution
Create a strategy

In order to construct a table of values, we will need to evaluate the function at the given values of x.

Apply the idea

Substituting x = -3, we have \begin{aligned} f\left(-3\right) & = \frac{-3}{3} - 5 \\ & = -6 \end{aligned}

Substituting x = 0, we have \begin{aligned} f\left(0\right) & = \frac{0}{3} - 5 \\ & = -5 \end{aligned}

Substituting x = 9, we have \begin{aligned} f\left(9\right) & = \frac{9}{3} - 5 \\ & = -2 \end{aligned}

Substituting x = 12, we have \begin{aligned} f\left(12\right) & = \frac{12}{3} - 5 \\ & = -1 \end{aligned}

Substituting x = 27, we have \begin{aligned} f\left(27\right) & = \frac{27}{3} - 5 \\ & = 4 \end{aligned}

A completed table for the function at the given values of x is

x-3091227
f\left( x \right)-6-5-2-14
b

Evaluate the function for f\left(2\right).

Worked Solution
Apply the idea

Substituting x =2, we have \begin{aligned} f\left(2\right) & = \frac{2}{3} - 5 \\ & = -\dfrac{13}{3} \end{aligned}

Example 2

Consider the graph:

-3
-2
-1
1
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5
x
-5
-4
-3
-2
-1
1
2
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5
f(x)
a

Evaluate the function for f\left(-1\right).

Worked Solution
Create a strategy

We can find the value of the function on the graph when x=-1 by moving down the y-axis until we intersect the curve.

Apply the idea

From x=-1, move down until we intersect the curve.

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x
-5
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f(x)

When x=-1, we can see on the graph that f\left(x\right)=-2.

b

Determine the value of x when f\left(x\right)=4.

Worked Solution
Create a strategy

To find the value of x on the graph where f\left(x\right)=4, move horizontally to the left from f\left(x\right)=4 until you intersect the curve, then move vertically downward until you reach the x-axis.

Apply the idea
-3
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x
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f(x)

When f\left(x\right)=4, we can see on the graph that x=-3.

Example 3

Consider the function f\left(x\right)=3x-5 to answer the following:

a

Find the range when the domain is \{-3,0,11\}.

Worked Solution
Create a strategy

We need to find the range (the values of f\left(x\right)) when the domain (the x-values) are \{-3,0,11\}. Substitute x-values and solve for f\left(x\right).

Apply the idea

Substitute x=-3 for x in the function.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(-3\right)\displaystyle =\displaystyle 3\left(-3\right)-5Substitute x=-3
\displaystyle {}\displaystyle =\displaystyle -9-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle -14Evaluate the subtraction

Substitute x=0 for x in the function.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(0\right)\displaystyle =\displaystyle 3\left(0\right)-5Substitute x=0
\displaystyle {}\displaystyle =\displaystyle 0-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle -5Evaluate the subtraction

Substitute x=11 for x in the function.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(11\right)\displaystyle =\displaystyle 3\left(11\right)-5Substitute x=11
\displaystyle {}\displaystyle =\displaystyle 33-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle 28Evaluate the subtraction

When the domain is \{-3,0,11\}, this function has a range of \{-14,-5,28\},

Reflect and check

This could be written in function notation as \{f\left(-3\right),\,f\left(0\right),\, f\left(11\right)\}=\{-14,\,-5,\,28\}.

b

Find the domain when the range is \{-2,4,7\}.

Worked Solution
Create a strategy

We need to find the domain (the x-values) when the range (the values of f\left(x\right) is \{-2,4,7\}. Substitute the f\left(x\right) values and solve for x.

Apply the idea

Replace f\left(x\right) with -2 in the equation and solve for x.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle -2\displaystyle =\displaystyle 3x-5Substitute f\left(x\right)=-2
\displaystyle 3\displaystyle =\displaystyle 3xAdd 5 to both sides
\displaystyle 1\displaystyle =\displaystyle xDivide both sides by 3

Replace f\left(x\right) with 4 in the equation and solve for x.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle 4\displaystyle =\displaystyle 3x-5Substitute f\left(x\right)=4
\displaystyle 9\displaystyle =\displaystyle 3xAdd 5 to both sides
\displaystyle 3\displaystyle =\displaystyle xDivide both sides by 3

Replace f\left(x\right) with 7 in the equation and solve for x.

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle 7\displaystyle =\displaystyle 3x-5Substitute f\left(x\right)=7
\displaystyle 12\displaystyle =\displaystyle 3xSubtract 5 from both sides
\displaystyle 4\displaystyle =\displaystyle xDivide both sides by 3

When this function has a range of \{-2,\,4,\,7\}, the domain is \{1,\,3,\,4\}.

Reflect and check

We can write this using function notation as \{f\left(-2\right),\,f\left(4\right),\, f\left(7\right)\}=\{1,\,3,\,4\}.

c

Evaluate f(7) - f(2)

Worked Solution
Create a strategy

Find the values of f\left(7\right) and f\left(2\right), then subtract the value of f\left(2\right) from the value of f\left(7\right).

Apply the idea

First, let's evaluate f\left(7\right).

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(7\right)\displaystyle =\displaystyle 3\left(7\right)-5Substitute x=7
\displaystyle {}\displaystyle =\displaystyle 21-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle 16Evaluate the subtraction

Now evaluate f(2).

\displaystyle f\left(x\right)\displaystyle =\displaystyle 3x-5Original function
\displaystyle f\left(2\right)\displaystyle =\displaystyle 3\left(2\right)-5Substitute x=2
\displaystyle {}\displaystyle =\displaystyle 6-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle 1Evaluate the subtraction

Now we can find f\left(7\right)-f\left(2\right) by substituting their values. f\left(7\right)-f\left(2\right)

=16-1

=15

Example 4

Let f\left( x \right) represent the height of a growing plant, f, in inches, where x represents the time since it was planted in days.

Plant Growth
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\text{Time (days)}
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\text{Height (inches)}
a

Interpret the meaning of f\left(10\right) = 8.

Worked Solution
Create a strategy

We can use the units of the given information and the graph to help with the interpretation.

Apply the idea

We're given that f\left( x \right) represents the height of a growing plant in inches, so to interpret f\left( 10 \right), we need to determine what an input of x=10 means. We know that x represents the time in days since the plant was planted. So this means that 10 days have passed since the plant was planted.

We also know that all of this is equal to 8. This is the output, or what our function f\left( x \right) is equal to. Since our function represents the height of a growing plant in inches, this means that our plant is 8 inches tall.

Based on the graph, when x=10, y=8 so f(10)=8 is represented by the ordered pair (10,8) on the graph.

The plant has a height of 8 inches 10 days after being planted.

b

Interpret the meaning of f\left(6\right).

Worked Solution
Apply the idea

We know that x represents the time in days since the plant was planted and x=6. So this means that 6 days have passed since the plant was planted.

Since f\left( x \right) represents the height of a growing plant in inches, f\left( 6 \right) represents the height of the plant 6 days after being planted.

Reflect and check

Using the graph, we can find the actual height of the plant after 6 days.

Plant Growth
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\text{Time (days)}
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\text{Height (inches)}
c

Interpret the meaning of f\left(x\right)=12.

Worked Solution
Apply the idea

We know that f\left( x \right) represents the height of a growing plant in inches, so if f\left( x \right)=12, then the height of the plant is 12 inches x days after being planted.

Reflect and check

By using the graph, we can find the number of days when the height of the plant is 12 inches.

Plant Growth
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\text{Time (days)}
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\text{Height (inches)}
Idea summary

An equation where the output variable is isolated like y=mx+b can be written as a function in the form, f\left(x\right)=mx+b. We evaluate a function, written in function notation as f\left(c\right), by replacing all values of x with c and evaluating the expression.

\displaystyle f\left(x\right)=y
\bm{f}
is the name of function
\bm{x}
is the input
\bm{y}
is the output

Outcomes

A.F.1

The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

A.F.1g

For any value, x, in the domain of f, determine f(x), and determine x given any value f(x) in the range of f, given an algebraic or graphical representation of a linear function.

A.F.2

The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

A.F.2g

For any value, x, in the domain of f, determine f(x) of a quadratic or exponential function. Determine x given any value f(x) in the range of f of a quadratic function. Explain the meaning of x and f(x) in context.

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