Exercises: Use function notation

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

A function $f$ is given by the table below.

$x$	$f(x)$
1	5
2	8
3	11
4	14

What is $f(3)$ ?

A.

3

B.

8

C.

11

D.

14

2.

Let $f(x) = 3x - 4$ . Evaluate $f(5)$ .

3.

The pool context: the number of swimmers $t$ hours after opening is $S(t)$ .
The statement " $S(3) = 85$ " means: "Three hours after opening, there are ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ swimmers."
In this context, the input $t = 3$ represents ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

number of swimmers:

what the input represents:

B

Fluency Practice

Evaluate each function as directed. Show your substitution steps.

1.

Let $f(x) = x^2 - 3x + 2$ . Evaluate $f(-1)$ .

2.

Let $g(t) = t^2 + 1$ . Evaluate $g\!\left(\dfrac{1}{2}\right)$ .

Express your answer as a fraction in simplest form.

3.

Let $h(x) = \sqrt{x - 3}$ . Which of the following is undefined (not in the domain)?

A.

$h(7)$

B.

$h(3)$

C.

$h(1)$

D.

$h(12)$

$Table showing four substitution steps for f(x) = 2x + 3 with inputs 4, a, a+1, and x+h$

4.

Let $f(x) = 2x + 3$ . Complete the table by filling in each output.

Input	Notation	Output
$x = 4$	$f(4)$	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
$x = a$	$f(a)$	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
$x = a+1$	$f(a+1)$	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
$x = x+h$	$f(x+h)$	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲

f(4):

f(a):

f(a+1):

f(x+h):

5.

Let $f(x) = x^2$ . Which statement is true when $a = 2$ and $b = 3$ ?

A.

$f(a + b) = f(a) + f(b)$

B.

$f(a + b) > f(a) + f(b)$

C.

$f(a + b) < f(a) + f(b)$

D.

$f(a + b) = f(a) \cdot f(b)$

C

Varied Practice

Use multiple representations — formulas, tables, and graphs — as directed.

D

Word Problems

Read each scenario carefully. Interpret function notation in terms of the context.

1.

A ride-share app charges a base fee plus a per-mile rate. The total fare in dollars for a trip of $d$ miles is modeled by $C(d) = 2.50d + 4$ .

1.

Evaluate $C(6)$ and interpret it in context.

What is the fare (in dollars) for a 6-mile trip?

2.

Solve $C(d) = 29$ to find the trip distance (in miles) that produces a $29 fare.

3.

Write a sentence interpreting the equation $C(d_1) = C(d_2)$ for two different trip distances $d_1 \neq d_2$ . Is this possible with this model? Explain.

2.

A city's population (in thousands) $t$ years after 2010 is modeled by $P(t) = 120 + 5t$ .

$P(0) = $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , which means "in the year ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , the population was ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ thousand."

P(0) value:

year:

population in thousands:

3.

The table below shows the height $h(t)$ (in feet) of a ball $t$ seconds after being thrown.

$t$ (seconds)	0	1	2	3	4
$h(t)$ (feet)	6	22	30	22	6

For how many values of $t$ in the table does $h(t) = 22$ ?

4.

Using the same ball table from the previous problem, evaluate $h(2)$ .

What is the maximum height of the ball (in feet)?

E

Error Analysis

Each problem shows a student's incorrect work. Identify the error and explain what went wrong.

$Error contrast card comparing Alex's incorrect f(3+4) = f(3)+f(4) = 25 with the correct f(7) = 49$

1.

Alex is given $f(x) = x^2$ and writes:
$f(3 + 4) = f(3) + f(4) = 9 + 16 = 25$

What mistake did Alex make?

A.

Alex evaluated $f(3)$ and $f(4)$ incorrectly.

B.

Alex incorrectly applied the distributive property to the function, treating $f$ as a multiplier instead of a rule.

C.

Alex should have added $3 + 4$ first, then squared, giving 49 — but the addition $9 + 16 = 25$ is still correct.

D.

Alex used the wrong function rule.

$Error contrast card showing Jordan mistakenly evaluating f(3)=5 versus correctly solving 2x-1=3 to get x=2$

2.

Jordan is asked: "Solve $f(x) = 3$ for $f(x) = 2x - 1$ ."
Jordan writes:
$f(3) = 2(3) - 1 = 5$
and concludes: "The answer is 5."

What is the error in Jordan's approach, and what is the correct solution?

A.

Jordan correctly solved the problem; the answer is $x = 5$ .

B.

Jordan evaluated $f(3)$ instead of solving $f(x) = 3$ . The correct answer is $x = 2$ .

C.

Jordan used the wrong arithmetic when evaluating $f(3)$ ; the correct evaluation gives $f(3) = 6$ .

D.

Jordan should have solved $f(x) = 5$ instead.

F

Challenge / Extension

These problems are bonus challenges. Show all steps.

1.

Let $f(x) = x^2 + 2x$ .

(a) Compute $f(x + h)$ and simplify your answer.
(b) Compute $f(x + h) - f(x)$ and simplify.
(c) Compute $\dfrac{f(x + h) - f(x)}{h}$ (assume $h \neq 0$ ) and simplify. What does this expression represent geometrically?

2.