3.2 Function Notation and Transformations
Key Takeaways
- Function notation means output for a given input; f(a) requires substituting a for every x before simplifying.
- Composition works from the inside out, so f(g(x)) means evaluate g first and then feed that output into f.
- Outside transformations change outputs directly, while inside transformations change inputs and often move horizontally in the opposite-looking direction.
- Domain restrictions should be checked before substitution, especially with denominators, even roots, logarithms, endpoints, and open dots on graphs.
Function Notation and Transformations
Function notation is compact, and that is why ACT Math likes it. The notation f(3) does not mean f times 3. It means the output of function f when the input is 3. Many missed questions come from treating notation like multiplication, applying transformations in the wrong direction, or ignoring domain restrictions before substituting.
The official ACT materials describe Functions questions as testing definition, notation, representation, application, translation, and graph features. That means you should expect the same idea to appear as an equation, table, graph, or word model. The skill is not memorizing one format; it is moving between formats quickly.
Evaluate Before You Simplify Too Much
When evaluating f(a), substitute a for every x in the function rule. Parentheses matter. If f(x) = 2x^2 - 3x and the input is -4, compute 2(-4)^2 - 3(-4) = 32 + 12 = 44. A common wrong path is 2(-16) + 12, which treats squaring as if it happens after multiplication by 2.
For an expression input such as f(x + 1), replace every x with (x + 1). If f(x) = x^2 - 5x, then f(x + 1) = (x + 1)^2 - 5(x + 1) = x^2 - 3x - 4. Keep the input grouped until expansion is complete.
| Notation | Meaning | Fast check |
|---|---|---|
| f(a) | plug a into f | Replace every x |
| f(x) = 7 | find input values with output 7 | Solve for x |
| f(g(x)) | do g first, then f | Work inside out |
| f^-1(x) | inverse function | Swap input and output |
| Domain | allowed x-values | Check denominators, roots, logs |
| Range | possible y-values | Read outputs or graph height |
Composition and Inverses
Composition means one function feeds another. In f(g(2)), start with g(2), then put that result into f. Do not average the functions, multiply their formulas, or work left to right just because f is written first. Parentheses decide the order.
Worked example: f(x) = x^2 + 1 and g(x) = 3x - 4. To find f(g(5)), compute g(5) = 11, then f(11) = 122. To find g(f(5)), compute f(5) = 26, then g(26) = 74. The order changes the answer.
An inverse reverses inputs and outputs. If f(2) = 9, then f^-1(9) = 2, provided f is one-to-one over the relevant domain. To find a simple inverse formula, write y = f(x), swap x and y, then solve for y. For f(x) = 4x - 7, write y = 4x - 7, swap to x = 4y - 7, and solve y = (x + 7)/4.
Transformations: Read Inside and Outside Separately
Transformations are predictable once you separate input changes from output changes. Outside changes move outputs; inside changes move inputs. The counterintuitive one is horizontal shift: f(x - h) moves the graph right h units because the input must be h larger to create the old output.
| New function | Graph change from y = f(x) |
|---|---|
| f(x) + k | up k |
| f(x) - k | down k |
| f(x - h) | right h |
| f(x + h) | left h |
| -f(x) | reflect across x-axis |
| f(-x) | reflect across y-axis |
| af(x), a > 1 | vertical stretch |
| f(ax), a > 1 | horizontal compression |
Worked example: g(x) = -f(x - 4) + 6. Start inside: x - 4 moves the graph right 4. The negative outside reflects it across the x-axis. The +6 moves it up 6. If a point (1, 3) is on f, then after the right shift it moves to (5, 3), after reflection to (5, -3), and after the upward shift to (5, 3). The final y-value happens to match the original here, but only because -3 + 6 = 3.
Domain Restrictions First
Domain is a frequent ACT trap because substitution can hide an illegal input. Denominators cannot equal zero. Even roots in real-number questions cannot have negative radicands. Logarithm inputs must be positive. Piecewise functions require choosing the correct interval before evaluating.
Use a two-pass routine. First, identify restrictions from the function rule. Second, apply the requested operation. For f(x) = 1/(x - 5), the input 5 is not allowed even if a later expression seems to cancel. For h(x) = sqrt(x + 2), the domain is x >= -2. For log(x - 3), the domain is x > 3.
When ACT Math provides a graph, domain and range become visual. Domain is how far the graph extends left and right. Range is how far it extends down and up. Closed dots include endpoints; open dots exclude endpoints. Arrows mean the graph continues, so do not invent an endpoint where the arrow appears.
Tables, Points, and Graph Features
A function can be tested without giving a formula. If a table says f(2) = 9, then the ordered pair (2, 9) is on the graph. If the question asks for f^-1(9), the answer is 2 because the inverse reverses that pair. If two different inputs lead to the same output, the inverse is not a function over that full table because one input would have to split into two outputs.
For transformations, mapping one point is often faster than redrawing a curve. Choose a clear point on the original graph, apply inside shifts to the x-coordinate, apply outside changes to the y-coordinate, and compare the result with the answer choices. This avoids vague visual language such as wider, lower, or flipped when the choices give exact coordinates.
If f(x) = 2x^2 - 5 and g(x) = x + 3, what is f(g(1))?
What is the domain of h(x) = sqrt(x - 4) + 2 in the real numbers?
The graph of g(x) = -f(x + 3) + 2 is made from the graph of f. Which description is correct?