Relations and functions

High School Math 1 · Unit 7 · function-foundations · Teacher edition

function-foundations

Name ______________________________Date ______________

1.
If $f(x) = 3x - 5$ , find $f(4)$ .
Answer: ______________
2.
Let $f(x) = \dfrac{x + 4}{2}$ . Find the value of $x$ for which $f(x) = 11$ .
Answer: ______________
3.
Which relation is a function?
1. (A)
  $\{(1, 2),\ (2, 3),\ (1, 4)\}$
2. (B)
  $\{(0, 1),\ (0, 2),\ (0, 3)\}$
3. (C)
  $\{(2, 1),\ (2, 3),\ (3, 4)\}$
4. (D)
  $\{(1, 2),\ (2, 2),\ (3, 2)\}$
4.
What is the domain of $f(x) = \sqrt{x - 1}$ , in interval notation?
1. (A)
  $(-\infty, \infty)$
2. (B)
  $[1, \infty)$
3. (C)
  $(1, \infty)$
4. (D)
  $[0, \infty)$
5.
The function $C(m) = 25 + 0.15m$ gives the cost in dollars of renting a van and driving it $m$ miles. For how many miles driven is the cost $61?
Answer: ______________

Answer key — Relations and functions

1.
7function-foundations-01
$f(4) = 3(4) - 5 = 12 - 5 = \mathbf{7}$ .
2.
18function-foundations-02
$\frac{x + 4}{2} = 11 \Rightarrow x + 4 = 22 \Rightarrow x = \mathbf{18}$ . Check: $f(18) = \frac{22}{2} = 11$ . ✓
3.
(D)
$\{(1, 2),\ (2, 2),\ (3, 2)\}$
function-foundations-03
In $\{(1, 2), (2, 2), (3, 2)\}$ the inputs $1, 2, 3$ are all different, so each has exactly one output — it is a function even though every output is $2$ . Answer: $\mathbf{\{(1,2),\ (2,2),\ (3,2)\}}$ . (The first set sends $1$ to both $2$ and $4$ ; the second sends $0$ to three outputs; the third sends $2$ to both $1$ and $3$ .)
4.
(B)
$[1, \infty)$
function-foundations-04
$x - 1 \ge 0 \Rightarrow x \ge 1$ , so the domain is $\mathbf{[1, \infty)}$ with a square bracket because $x = 1$ gives $\sqrt{0} = 0$ . (All reals ignores the root; $(1, \infty)$ wrongly excludes $1$ ; $[0, \infty)$ forgets the $-1$ inside.)
5.
240function-foundations-05
$25 + 0.15m = 61 \Rightarrow 0.15m = 36 \Rightarrow m = \frac{36}{0.15} = \mathbf{240}$ miles. Check: $25 + 0.15(240) = 25 + 36 = 61$ . ✓