Topics
5. Functions & Linear Relationships
topic badge
United States of AmericaTennessee
Integrated Math 1

5.01 Evaluating functions

Lesson
Practice
Lesson

Concept summary

A mathematical relation is a mapping from a set of input values, called the domain, to a set of output values, called the range. A relation can also be described as a set of input-output pairs.

Input

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

Output

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

Relations in general have no further restrictions than mapping domain elements to range elements. By adding the restriction that each input value maps to exactly one output value, we define a particularly useful type of relation, called a function.

Function

A relation for which each element of the domain corresponds to exactly one element of the range

Functions are usually written using a particular notation called function notation: for a function f when x is a member of the domain, the symbol f\left(x\right) denotes the corresponding member of the range.

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

Worked examples

Example 1

Consider the equation x - 3y = 15 where x is the independent variable.

a

Rewrite the equation using function notation.

Approach

Since x is the independent variable, we want to rearrange the equation to isolate y, and then replace y with function notation. We can choose a symbol to represent the function, such as f.

Solution

Rearranging the equation:

\displaystyle -3y\displaystyle =\displaystyle -x + 15Subtract x from both sides
\displaystyle y\displaystyle =\displaystyle \frac{x}{3} - 5Divide both sides by -3

We can now rewrite the equation using function notation as f\left(x\right) = \frac{x}{3} - 5

b

Evaluate the function when x = 9.

Solution

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

Example 2

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.

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

Approach

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

Solution

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

Outcomes

M1.N.Q.A.1

Use units as a way to understand real-world problems.*

M1.N.Q.A.1.B

Use appropriate quantities in formulas, converting units as necessary.*

M1.F.IF.A.1

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

M1.F.IF.A.2

Use function notation.*

M1.F.IF.A.2.A

Use function notation to evaluate functions for inputs in their domains, including functions of two variables.

M1.F.IF.A.2.B

Interpret statements that use function notation in terms of a context.

M1.F.IF.A.3

Understand geometric formulas as functions.*

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP6

Attend to precision.

M1.MP8

Look for and express regularity in repeated reasoning.

What is Mathspace

About Mathspace