Exercises: Understand function definition

A

Recall / Warm-Up

These problems review skills from Grade 8.

1.

Which statement best describes what a function does?

A.

It assigns each input exactly one output.

B.

It assigns each output exactly one input.

C.

It pairs inputs and outputs so that every output is used.

D.

It creates a formula relating two variables.

2.

A table shows these ordered pairs: (1, 4), (2, 7), (3, 4), (4, 9). Does this table represent a function?

A.

Yes, because each input has exactly one output.

B.

No, because the output 4 appears twice.

C.

No, because the inputs are not consecutive even numbers.

D.

Yes, but only if a formula can be written for it.

3.

A function is defined by the set of ordered pairs {(0, 1), (1, 3), (2, 5), (3, 7)}. What is the range of this function?

A.

{0, 1, 2, 3}

B.

{1, 3, 5, 7}

C.

All real numbers

D.

{0, 1, 2, 3, 5, 7}

B

Fluency Practice

Evaluate each function at the given input.

1.

Let $f(x) = 2x + 5$ . Find $f(3)$ .

2.

Let $g(x) = x^2 - 4$ . Find $g(5)$ .

3.

Let $h(t) = 3t - 7$ . Find $h(-2)$ .

4.

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

5.

Let $p(n) = \frac{n+1}{2}$ . Find $p(7)$ .

C

Varied Practice

Apply your understanding of functions in different contexts.

$x$	1	2	3	4	5
$f(x)$	6	9	9	15	20

D

Word Problems

Apply function concepts to real-world scenarios.

1.

A phone plan charges a monthly fee. The total cost in dollars after $m$ months is given by $C(m) = 45m + 30$ , where the 30 represents an activation fee.

What is the total cost after 6 months? Give your answer in dollars.

2.

A ball is thrown upward from a rooftop. Its height in feet above the ground after $t$ seconds is modeled by $h(t) = -16t^2 + 32t + 48$ .

1.

What is the height of the ball at $t = 0$ seconds?

2.

What is the height of the ball at $t = 1$ second?

3.

A taxi company charges a fare based on distance. The cost in dollars for a ride of $d$ miles is $C(d) = 2.50d + 3.00$ , where the domain is $0 \le d \le 40$ .

What is the maximum cost of a taxi ride under this model? Give your answer in dollars.

E

Error Analysis

Each problem shows student work with an error. Identify the mistake.

1.

Given $f(x) = x^2 + 3$ , a student wrote:

$f(5) = f \times 5 = 5 \times 5 + 3 \times 5 = 40$

What error did the student make?

A.

The student treated $f(x)$ as $f$ times $x$ instead of evaluating the function at $x = 5$ .

B.

The student forgot to square the 5.

C.

The student used the wrong function rule.

D.

The student made an arithmetic error in the final step.

2.

A student was asked for the domain and range of the function $f(x) = \sqrt{x}$ and wrote:

Domain: all real numbers. Range: all real numbers.

What is wrong with the student's answer?

A.

The domain should be $x \ge 0$ because you cannot take the square root of a negative number.

B.

The range is correct but the domain should be $x > 0$ .

C.

The domain is correct but the range should be $y > 0$ .

D.

Both the domain and range should be all integers.

F

Challenge / Extension

These problems go beyond the basics. Take your time and show your reasoning.

1.

Let $f(x) = 2x - 1$ . Find $f(f(3))$ . Show your work and explain what $f(f(3))$ means.

2.

Let $f(x) = x^2$ with domain $\{-3, -2, -1, 0, 1, 2, 3\}$ . What is the range of $f$ ?

A.

{-9, -4, -1, 0, 1, 4, 9}

B.

{0, 1, 4, 9}

C.

All non-negative real numbers