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.

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.

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

b

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

Worked Solution

Consider the graph:

a

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

Worked Solution

b

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

Worked Solution

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

b

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

Worked Solution

c

Evaluate f(7) - f(2)

Worked Solution

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