Absolute Value, Piecewise & Step Functions — Practice
Some functions change their rule partway along. Work out where the rule changes, then treat each piece separately.
What these questions cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Absolute value — |x| — distance from zero, always positive or zero.
- Isolate — Get the variable alone on one side.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Parabola — The U-shaped graph of a quadratic function.
Problem 1
Evaluate |-9|.
Worked answer
9
|-9| = 9
Problem 2
Evaluate |4 - 11|.
Worked answer
7
4 - 11 = -7
|-7| = 7
Problem 3
Solve |x| = 6.
Worked answer
6 and -6
x = 6 or x = -6
Problem 4
Find the vertex of y = |x - 5|.
Worked answer
(5, 0)
x - 5 = 0 when x = 5
y = 0 there
Problem 5
For f(x) = x + 1 if x < 2 and f(x) = 3x if x ≥ 2, find f(0).
Worked answer
1
f(0) = 0 + 1 = 1
Problem 6
Solve |x - 3| = 8.
Worked answer
11 and -5
x - 3 = 8 or x - 3 = -8
x = 11 or x = -5
Problem 7
Solve |2x + 1| = 9.
Worked answer
4 and -5
2x + 1 = 9 gives x = 4
2x + 1 = -9 gives x = -5
Problem 8
Solve 2|x| + 5 = 17.
Worked answer
6 and -6
2|x| = 12
|x| = 6
x = ± 6
Problem 9
Solve |x + 4| = -3. If there is no solution, answer "none".
Worked answer
none
Absolute value is never negative
So no solution
Problem 10
For f(x) = x2 if x ≤ 1 and f(x) = 2x + 3 if x > 1, find f(4).
Worked answer
11
f(4) = 2(4) + 3 = 11
Problem 11
The graph of y = -|x| + 2 has a vertex at (0, 2) and
- opens downward
- opens upward
- is a straight line
- is a parabola
Worked answer
opens downward
-|x| turns the V upside down
So (0, 2) is a maximum
Problem 12
Find the vertex of y = |x + 7| - 4.
Worked answer
(-7, -4)
x + 7 = 0 gives x = -7
y = -4 there
Problem 13
Solve 3|x - 1| - 5 = 10.
Worked answer
6 and -4
3|x-1| = 15
|x - 1| = 5
x = 6 or x = -4
Problem 14
A garage charges $5 for the first hour and $3 for each hour after. What does 4 hours cost?
Worked answer
14
5 + 3(3)
= 5 + 9 = $14
Problem 15
For f(x) = -x if x < 0 and f(x) = x2 if x ≥ 0, find f(-3) + f(3).
Worked answer
12
f(-3) = 3 from the first rule
f(3) = 9 from the second
3 + 9 = 12
Problem 16
Find the vertex of y = |3x - 12| + 1.
Worked answer
(4, 1)
3x - 12 = 0 gives x = 4
y = 1 there