TBIL-Precal Function Notation (FN2)

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Section 2.2 Function Notation (FN2)

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Objectives

Use and interpret function notation to evaluate a function for a given input value and find a corresponding input value given an output value.

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Subsection 2.2.1 Activities

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Remark 2.2.1.

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As we saw in the last section, we can represent functions in many ways, like using a set of ordered pairs, a graph, a description, or an equation. When describing a function with an equation, we will often use function notation.

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y

is written as a function of

x,

like in the equation

y = x + 5,

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we can replace the

y

with

f (x)

and get the function notation

f (x) = x + 5 .

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The

x

is the input variable, and

f (x)

is the

y

-value or output that corresponds to

x .

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Generally, we use the letter

f

for functions. Other letters are okay as well;

g (x)

and

h (x)

are common. If we are using multiple functions at one time, we often denote them with different letters so we can refer to one without any confusion as to which function we mean.

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Activity 2.2.2.

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Rewrite the following equations using function notation. In each case, assume

y

is a function of the variable

x .

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(a)

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y = 2 x + 14

Answer.

f (x) = 2 x + 14

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(b)

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y + x = 3 x^{2} - 5

Answer.

f (x) = 3 x^{2} - x - 5

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(c)

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\frac{2}{x} - x^{4} = y - 5

Answer.

f (x) = \frac{2}{x} - x^{4} + 5

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Activity 2.2.3.

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Let

f (x) = 3 x^{2} - 4 x + 1 .

Find the value of

f (x)

for the given values of

x .

Table 2.2.4.

$x$	$f (x)$
$- 5$
$- \frac{1}{2}$
$0$
$2$
$10$

Answer.

$x$	$f (x)$
$- 5$	$96$
$- \frac{1}{2}$	$\frac{15}{4}$
$0$	$1$
$2$	$5$
$10$	$261$

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Remark 2.2.5.

If we are asked to find the value of

f (x)

for a certain

x

-value, say

x = 5,

we use the notation

f (5)

to indicate that.

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Activity 2.2.6.

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Let

f (x),

g (x),

and

h (x)

be defined as shown.

\begin{aligned} f (x) & = 3 x^{2} - 4 x + 1 \\ g (x) & = \sqrt{13 - x^{2}} \\ h (x) & = \frac{x^{2} - 6 x + 8}{x^{2} - 4 x + 3} \end{aligned}

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Find the following, if they exist.

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(a)

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f (- 4),

f (0),

and

f (2)

Answer.

f (- 4) = 65,

f (0) = 1,

and

f (2) = 5

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(b)

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g (0),

g (2),

and

g (8)

Answer.

g (0) = \sqrt{13},

g (2) = 3,

and

g (8)

is not defined

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(c)

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h (3),

h (4),

and

h (10)

Answer.

h (3)

is not defined,

h (4) = 0,

and

h (10) = \frac{16}{21}

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Remark 2.2.7.

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Sometimes functions are made up of multiple functions put together. We call these piecewise functions. Each piece is defined for only a certain interval, and these intervals do not overlap. When evaluating a piecewise function at a given

x

-value, we first need to find the interval that includes the

x

-value, and then plug in to the corresponding function piece.

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Activity 2.2.8.

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Let

f (x)

be a piecewise function as shown below.

f (x) = {\begin{cases} x^{2} + 3, & x < 5 \\ 9 - 2 x, & x \geq 5 \end{cases}

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(a)

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On which interval from the piecewise function does the value

x = 1

belong?

$x < 5$
$x \leq 5$
$x > 5$
$x \geq 5$

Answer.

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(b)

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Find

f (1) .

Answer.

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(c)

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On which interval from the piecewise function does the value

x = 5

belong?

$x < 5$
$x \leq 5$
$x > 5$
$x \geq 5$

Answer.

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(d)

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Find

f (5) .

$- 10$
$- 5$
$- 1$
$17$
$28$

Answer.

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Remark 2.2.9.

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We’ve been practicing evaluating functions at specific numeric values. It’s also possible to evaluate a function given an expression involving variables.

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