Function Concepts • Topic 4 of 5

Range

The range of a function is the set of all output (y) values it can produce. For f(x) = x² + k, the smallest value of x² is 0, so the outputs start at k and go up: the range is y ≥ k. The same reasoning works for f(x) = |x| + k, since |x| ≥ 0. A negative coefficient flips this: f(x) = −x² + k has a maximum of k, so the range is y ≤ k. Finding a range usually means finding the minimum or maximum output and noting which direction the function extends. The SAT tests this mainly for squared and absolute-value functions shifted up or down.

sat29t4 graphO(0, 4)y ≥ 4Rangey = x² + 4 has minimum 4, so the range is y ≥ 4

✅ Solved examples

1. Find the range of f(x) = x² + 4.
x² ≥ 0, so the minimum is 4: range y ≥ 4.
2. Find the range of f(x) = |x| − 1.
|x| ≥ 0, so minimum −1: range y ≥ −1.
3. Find the range of f(x) = x² − 3.
Minimum −3: range y ≥ −3.
4. Find the range of f(x) = −x² + 5.
Maximum 5: range y ≤ 5.

✏️ Practice — try these, take hints as needed

1. Find the range of f(x) = x² + 2.
x² ≥ 0.
Minimum is 2.
—
y ≥ 2.
2. Find the range of f(x) = |x| + 3.
|x| ≥ 0.
Minimum 3.
—
y ≥ 3.
3. Find the range of f(x) = x² − 7.
Minimum −7.
—
—
y ≥ −7.
4. Find the range of f(x) = x².
Smallest output.
x² ≥ 0.
—
y ≥ 0.
5. Find the range of f(x) = −x² + 2.
Opens down; maximum 2.
—
—
y ≤ 2.

📝 Topic test — 8 questions

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