Absolute Value, Piecewise & Step Functions — Challenge
Some functions change their rule partway along. Work out where the rule changes, then treat each piece separately.
What these problems 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
How does the graph of y=f(x)+3 compare with the graph of y=f(x)?
- Shifted up 3
- Shifted down 3
- Shifted right 3
- Shifted left 3
- Stretched by 3
Worked answer
Shifted up 3
Adding 3 to the output raises every point of the graph by 3.
So the whole graph moves up 3 units, with its shape unchanged.
Changes applied outside the function affect the output and move the graph vertically.
Problem 2
The graph of y=(x-2)2 is the graph of y=x2 shifted which way?
- Left 2
- Right 2
- Up 2
- Down 2
- Reflected
Worked answer
Right 2
The vertex of y = x2 is at x=0; the vertex of y = (x-2)2 is where the bracket is zero, at x=2.
So the graph has moved 2 units right, even though the sign inside the bracket is negative.
Changes applied to the input move the graph horizontally, and in the opposite direction to the sign.
Problem 3
If f(x)=2x+1, what is f-1(5)?
- 5
- 2
- 11
- 3
- 1
Worked answer
2
The inverse answers the question: which input gives the output 5?
Solve 2x+1 = 5: that gives 2x = 4 and x = 2.
So f-1(5) = 2. Substituting 5 into f instead would give 11, which is the trap.
Problem 4
The graph of y=-f(x) is a reflection of y=f(x) in what?
- The y-axis
- The x-axis
- The origin
- The line y=x
- It is not a reflection
Worked answer
The x-axis
Negating the output changes the sign of every y value while leaving each x alone.
A point above the x-axis moves to the mirror position below it, so the reflection is in the x-axis.
Writing y = f(-x) instead would negate the input and reflect in the y-axis.
Problem 5
If f(x)=x2 and g(x)=x+1, what is f(g(2))?
- 5
- 7
- 9
- 10
- 4
Worked answer
9
Work from the inside outwards: g(2) = 2+1 = 3.
Then apply f: f(3) = 32 = 9.
The other order, g(f(2)), gives g(4) = 5, which is why 5 appears among the options.
Problem 6
What is the domain of f(x)=√x-4?
- x>4
- x≤ 4
- x≥ 4
- All real numbers
- x≥ 0
Worked answer
x≥ 4
A square root of a negative number is not a real number, so the inside must be at least zero.
x-4≥ 0 gives x≥ 4.
Equality is allowed, because √0 = 0 is perfectly well defined, which is why the answer is not x>4.
Problem 7
The graph of y=3f(x) is which change to the graph of y=f(x)?
- A vertical compression by factor 3
- A vertical stretch by factor 3
- A shift up 3
- A shift right 3
- A reflection
Worked answer
A vertical stretch by factor 3
Multiplying the output by 3 triples the distance of every point from the x-axis.
So the graph is stretched vertically by a factor of 3; points on the x-axis stay put, since 3· 0 = 0.
A factor between 0 and 1 would compress it instead.
Problem 8
Is f(x)=x3 increasing or decreasing?
- Increasing everywhere
- Decreasing everywhere
- Increasing then decreasing
- Constant
- Decreasing then increasing
Worked answer
Increasing everywhere
Cubing preserves order: if a < b then a3 < b3, whether the numbers are positive or negative.
For example -2 < -1 and -8 < -1, and 1 < 2 with 1 < 8.
So the function increases across the whole real line, unlike x2, which falls and then rises.