Chapter 2 (Input/Output, Domain/Range, Piecewise, Inverses, Concavity)

Question 1

Sketch a function that is constant over the domain {-3 ≤ x≤ 2}

Answer 1

This will be a horizontal segment drawn from x = -3 to x = 2.

Question 2

If H is a function of W, which is the input?

Answer 2

W is the input (and the independent variable)

Question 3

Imagine a table of values for two variables, x and y, and two ordered pairs are (2, 4) and (2, 6). Can y be a function of x?

Answer 3

NO. There are two different outputs (y’s) for the same input (x = 2).

Question 4

If T(w) = 3w +5, T(h) = 14 and d = h – 2, find d.

Answer 4

d = 1. First, solve T(w) for when T(h) = 14. This gives h = 3. Then subtract 2 to get d = 1.

Question 5

If N(t) = 10 + 2t, find N(g).

Answer 5

10 + 2g

Question 6

If T(y) = 8y – 5, find T(n + 3)

Answer 6

8(n + 3) – 5 = 8n + 24 – 5 = 8n + 19

Question 7

Describe the domain of any quadratic function.

Answer 7

{all reals}

Question 8

Describe the range of any quadratic function.

Answer 8

It will be either be {all reals ≤ Y} or {all reals ≤ Y}, where Y is the y-coordinate of the vertex of the parabola.

Question 9

What is the domain of y = (x + 5) ÷ (x – 3) ?

Answer 9

{all reals ≠ 3}

Question 10

What is the one exception in the range of f(x) = 2x ÷ (x – 3) ?

Answer 10

2 (i.e. the output, y, will never be 2)

Question 11

What is the domain of g(x) = √(x + 6) ?

Answer 11

{all reals ≥ -6}

Question 12

Assume your oxygen intake is a function of your heart rate. What would be a reasonable domain for this function?

Answer 12

INTEGERS between 50 and 140.

Question 13

Given this piecewise function, find f(1).

2x + 6 for {0 ≤ x ≤ 2}

f(x) = -6 for {2 < x < 5}

12 – 4x for {5 ≤ x ≤ 8}

Answer 13

f(1) = 2(1) + 6 = 8 (Hint: the correct formula depends upon which domain your input falls)

Question 14

Given this piecewise function, find f(5).

2x + 6 for {0 ≤ x ≤ 2}

f(x) = -6 for {2 < x < 5}

12 – 4x for {5 ≤ x ≤ 8}

Answer 14

f((5) = 12 – 4(5) = 12 – 20 = -8 (Hint: the correct formula depends upon which domain your input falls)

Question 15

Given this piecewise function, find f(9).

2x + 6 for {0 ≤ x ≤ 2}

f(x) = -6 for {2 < x < 5}

12 – 4x for {5 ≤ x ≤ 8}

Answer 15

Undefined. The input 9 does not fall into any of the given domains.

Question 16

Graph this piecewise function:

2x + 6 for {0 ≤ x ≤ 2}

f(x) = -6 for {2 < x < 5}

12 – 4x for {5 ≤ x ≤ 8}

Answer 16

First piece: a positively-sloped segment from x=0 to x=2, with CLOSED circles on each end. Second piece: a horizontal segment at y = -6, from x=2 to x=5, with OPEN circles on each end. Third piece: a negatively-sloped segment from x=5 to x=8, with CLOSED circles on each end.