Review of Functions

Show your work for each problem.

1.
If , what is ?
1. a)10
2. b)12
3. c)9
4. d)14
Answer: 14
2.
If , find .
Answer: 8
3.
Which graph passes the vertical line test?
1. a)A horizontal parabola
2. b)A circle
3. c)A parabola opening upward
4. d)A figure-eight
Answer: A parabola opening upward
A parabola opening upward passes the vertical line test because each vertical line crosses it at most once. Circles and horizontal parabolas fail because vertical lines can cross them twice.
4.
If , find .
Answer: -11
1. Substitute into the expression. What is ? -6
2. Now subtract 5 from your answer. What is ? -11
5.
If , find .
Answer: 2
. The absolute value makes any negative result positive.
6.
What is the domain of ?
1. a)
2. b)
3. c)
4. d)All real numbers
Answer:
For to be defined, we need , which gives .
7.
Given and , find .
Answer: 18
1. First, find . What is ? 6
2. Now find . What is ? 12
3. Add your results: 18
8.
Given and , find .
Answer: 9
9.
Given and , find .
Answer: 7
1. In , which function do we apply first? f
2. Find . What is ? 10
3. Now apply g to your result. What is ? 7
10.
If , what value is NOT in the domain?
1. a)
2. b)
3. c)
4. d)
Answer:
When , the denominator becomes , and division by zero is undefined. So is not in the domain.
11.
Given and , find .
Answer: 48
12.
Evaluate the piecewise function at :
Answer: 3
1. Is less than 0 or greater than or equal to 0? less than
2. Which piece should we use? first
3. Calculate 3
13.
What is the range of ?
1. a)All real numbers
2. b)
3. c)
4. d)
Answer:
Since for all real , the minimum value of is . Therefore, the range is or .
14.
Given and , find .
Answer: 1
1. Find 5
2. Find 5
3. Calculate 1
15.
If and , find .
Answer: 3
. Setting : , so . Check: , then ✓
16.
Find the domain of .
Answer: [0, 4) U (4, infinity)
1. For the square root, what must be true about x? x >= 0
2. What value makes the denominator zero? 4
3. Combining both restrictions: x must be at least 0, but cannot be what value? 4
17.
If and , which expression equals ?
1. a)
2. b)
3. c)
4. d)
Answer:
. Note: would be different!
18.
If , find .
Answer: 2
1. First, find 3
2. Now find 2
3. So 2