Functions
Function notation, domain and range, transformations and reading function graphs.
What it tests
Functions questions test whether you can use function notation, find domain and range, read function graphs and tables, and understand how changing an equation shifts or flips its graph. You’ll work with linear, quadratic, exponential, and piecewise functions.
How it shows up
- The Math test is 45 questions in 50 minutes — a little over a minute each. Every question has four answer choices, a calculator is allowed on the whole section, there is no penalty for guessing, and no formula sheet is given.
- Functions is one of the larger Preparing for Higher Math categories.
- Typical questions: “What is f(−2)?”, “What is f(g(3))?”, a graph with “Which of the following is true?”, a table with “Which function fits?”, and transformations like “the graph of y = f(x − 3)”.
Key rules & concepts
The content you actually need to know.
Function notation
- f(3) means: replace every x with 3. Use parentheses — f(−2) with x2 is (−2)2 = 4.
- f(a + 1): replace x with the whole expression (a + 1).
- Composition f(g(x)): work inside-out — find g(x) first, then plug that into f.
Domain and range
- Domain = all allowed inputs. Exclude values that make a denominator 0 or put a negative under an even root.
- Range = all possible outputs. For y = x2, the range is y ≥ 0.
Reading graphs and tables
- f(a) is the y-value of the graph at x = a.
- Zeros / x-intercepts are where f(x) = 0. The y-intercept is f(0).
- A table with equal steps in x: constant differences in y → linear; constant ratios → exponential.
- If f(r) = 0 for a polynomial, then (x − r) is a factor.
Transformations
- f(x) + k shifts up k; f(x) − k shifts down k.
- f(x − h) shifts RIGHT h; f(x + h) shifts LEFT h (opposite of what the sign suggests).
- −f(x) reflects over the x-axis; f(−x) reflects over the y-axis.
- a · f(x) stretches vertically by a factor of a (shrinks if 0 < a < 1).
Inverse and piecewise functions
- Inverse: write y = f(x), swap x and y, and solve for y. f(f−1(x)) = x.
- Piecewise: first decide which interval the input falls in, then use only that piece’s rule.
Strategy & traps
Step by step
- 1Say the notation in words: “f(2) is the output when the input is 2.”
- 2Substitute with parentheses around every input.
- 3For graphs, read the axis labels, then find the exact point the question asks about.
- 4For transformations, test one easy point (like the vertex) to see where it moves.
Shifting the wrong way
y = (x − 3)2 moves the parabola RIGHT 3, not left.
Composition in the wrong order
f(g(x)) means g first. f(g(2)) and g(f(2)) are usually different.
Squaring a negative without parentheses
(−2)2 = 4, but −22 = −4. Always plug in with parentheses.
Worked examples
Try each one first, then reveal the solution.
Example 1
If f(x) = 2x2 − 3x + 1, what is f(−2)?
- A−1
- B3
- C15
- D17
Example 2
Let f(x) = x + 4 and g(x) = 3x2. What is f(g(2))?
- A16
- B18
- C36
- D108
Example 3
The graph of y = x2 is shifted 3 units to the left and 2 units down. Which equation describes the new graph?
- Ay = (x + 3)2 − 2
- By = (x − 3)2 − 2
- Cy = (x + 3)2 + 2
- Dy = (x − 2)2 + 3
Practice
Pick an answer for instant feedback. Answers lock once chosen.
Question 1
What is the domain of g(x) = √(x − 5) / (x − 9)?
Question 2
If f(x) = (x + 7)/2, which of the following is f−1(x)?
Question 3
The function f is defined by f(x) = x2 − 1 when x < 2, and f(x) = 3x + 4 when x ≥ 2. What is f(2) + f(−3)?
Question 4
Values of f(x)
| x | f(x) |
|---|---|
| 0 | 7 |
| 1 | 4 |
| 2 | 1 |
| 3 | −2 |
Which function is consistent with the table?
Question 5
A quadratic function f has a leading coefficient of 2, and its graph crosses the x-axis only at x = −1 and x = 3. Which could be f(x)?