What does f(x) mean on the ACT?

f(x) is function notation naming a rule f applied to an input x; it is not f multiplied by x. To evaluate f(x) at a number, substitute that number wherever x appears and simplify. For example, if f(x) = x + 4 then f(2) = 6.

How do function graph transformations work?

Changes outside the function move the graph vertically as expected: f(x) + k shifts up k. Changes inside move it horizontally and opposite the sign: f(x - h) shifts right h. A negative outside, -f(x), reflects over the x-axis; a negative inside, f(-x), reflects over the y-axis.

Why does f(x - 3) shift the graph to the right?

Because the change is inside the function, you must feed in an x that is 3 larger to reproduce the same output the original had, so every point moves 3 units to the right. Inside changes always act opposite to their sign.

How do you compose functions like f(g(x))?

Work from the inside out. First evaluate the inner function g at the input, then use that result as the input to the outer function f. For example, if f(x) = x^2 and g(x) = x + 1, then f(g(2)) = f(3) = 9.

CramClub • Functions and Graph Transformations

$f(x)$ = output of rule $f$ at input $x$

$f(x) + k$ moves up $k$

$f(x - h)$ moves right $h$

$-f(x)$ over $x$ -axis; $f(-x)$ over $y$ -axis

Function Notation Is a Machine

A function is a rule that assigns exactly one output to each input. The notation $f(x)$ names the rule $f$ and the input $x$ ; it does not mean $f$ times $x$ . Think of $f$ as a machine: you feed in a number, it applies its rule, and one number comes out. On the ACT you will evaluate functions, read them off graphs and tables, compose them, and — most importantly — transform their graphs. These skills recur throughout the higher-math half of the test.

Evaluating and Composing

To evaluate a function, substitute the input everywhere $x$ appears. If $f(x) = x^2 - 3x + 1$ , then

$f(4) = 4^2 - 3(4) + 1 = 16 - 12 + 1 = 5$

You can also feed a function an expression. With $g(x) = 2x + 5$ , we get $g(a + 1) = 2(a + 1) + 5 = 2a + 7$ .

Composition feeds one function's output into another. The notation $f(g(x))$ means "do $g$ first, then $f$ ." If $f(x) = x^2$ and $g(x) = x + 3$ , then

$f(g(2)) = f(2 + 3) = f(5) = 25$

Order matters: $g(f(2)) = g(4) = 7$ is different.

Domain and Range

The domain is the set of allowed inputs; the range is the set of possible outputs. On the ACT, two situations restrict the domain: you cannot divide by zero, and you cannot take the square root of a negative. For $h(x) = \sqrt{x - 2}$ , the inside must be $\ge 0$ , so the domain is $x \ge 2$ .

The Four Transformations

Transforming a graph is the ACT's favorite function topic. Start with a base graph $y = f(x)$ and change it in one of four ways. The key idea: changes outside the function (added to or multiplying $f(x)$ ) move the graph vertically and behave "as expected," while changes inside the function (attached to $x$ ) move it horizontally and behave "backwards."

Transformation	Effect on the graph
$f(x) + k$	Shift up $k$ (down if $k < 0$ )
$f(x - h)$	Shift right $h$ (left if $h < 0$ ) — opposite of the sign
$-f(x)$	Reflect over the $x$ -axis
$f(-x)$	Reflect over the $y$ -axis
$a \cdot f(x)$	Vertical stretch if $a > 1$ , compress if $0 < a < 1$

The most tested — and most misread — is the horizontal shift. In $f(x - 3)$ the graph moves right 3, even though the sign is negative, because you must input a value 3 larger to get the same output.

Combining Transformations

Real ACT questions stack transformations. Read $g(x) = -f(x + 2) - 1$ piece by piece:

$f(x + 2)$ : shift left 2 (inside, opposite sign),
the leading $-$ : reflect over the $x$ -axis,
the $-1$ : shift down 1.

So the base graph is flipped upside down, slid two units left, and dropped one unit.

Reading Graphs and Tables

The ACT often gives $f$ as a graph or table instead of a formula and asks for a value or a solution. To find $f(3)$ from a graph, go to $x = 3$ and read the height. To solve $f(x) = 0$ , find where the graph crosses the $x$ -axis (the zeros). To solve $f(x) = g(x)$ , find where two graphs intersect.

A Reliable Game Plan

Evaluate? Substitute the input everywhere and simplify.
Compose $f(g(x))$ ? Work the inner function first.
Vertical change (outside $f$ )? It behaves normally: $+k$ up, $-k$ down.
Horizontal change (inside $f$ )? It behaves backwards: $f(x - h)$ moves right $h$ .
Negative sign? Outside flips vertically; inside flips horizontally.

Common Mistakes to Avoid

Reading $f(x)$ as multiplication instead of "the function evaluated at $x$ ."
Shifting $f(x - 4)$ left instead of right — inside changes are the opposite of their sign.
Doing composition in the wrong order: $f(g(x))$ means $g$ first.
Confusing a reflection over the $x$ -axis, $-f(x)$ , with one over the $y$ -axis, $f(-x)$ .

If $f(x) = 3x^2 - x + 2$ , find $f(-2)$ .

The graph of $y = f(x)$ is moved to produce $y = f(x) + 4$ . Describe the change.

If $f(x) = x + 5$ and $g(x) = x^2$ , what is $f(g(3))$ ?

The graph of $y = f(x)$ passes through the point $(2, 5)$ . Through which point does $y = f(x - 3)$ pass?

The graph of $y = f(x)$ has a lowest point (minimum) at $(1, -2)$ . Where is the corresponding point on $y = -f(x) + 3$ ?

A function $f$ has $f(1) = 4$ , $f(2) = 1$ , $f(3) = 6$ , and $f(4) = 2$ . If $g(x) = f(x) - 1$ , for what value of $x$ does $g(x) = 5$ ?

Reading $f(x)$ as $f$ times $x$ and "distributing," e.g. treating $f(3)$ as $3f$ .

Function notation names a rule: $f(3)$ means substitute 3 into the rule for $f$ . It is not multiplication.

Shifting $f(x - 5)$ to the left because the sign is negative.

Inside changes move opposite the sign: $f(x - 5)$ shifts the graph right 5. Test a point — the input must grow to reproduce the old output.

Composing in the wrong order, computing $g(f(x))$ when the question asks for $f(g(x))$ .

In $f(g(x))$ , $g$ is the inner function and runs first. Work from the inside out, one function at a time.

Mixing up the two reflections, applying $-f(x)$ when a $y$ -axis flip $f(-x)$ is wanted.

A minus sign outside ( $-f(x)$ ) flips over the $x$ -axis; a minus sign inside ( $f(-x)$ ) flips over the $y$ -axis.

If $f(x) = 2x + 7$ , what is $f(5)$ ?

The graph of $y = f(x)$ is shifted to form $y = f(x) - 6$ . How does the graph change?

If $f(x) = x^2$ and $g(x) = x - 4$ , what is $f(g(6))$ ?

The point $(4, 9)$ lies on the graph of $y = f(x)$ . Which point must lie on the graph of $y = f(x + 2)$ ?

The graph of $y = f(x)$ has a maximum at $(3, 5)$ . What are the coordinates of the corresponding point on $y = -f(x)$ ?

A function satisfies $f(0) = 3$ , $f(1) = 7$ , and $f(2) = 3$ . If $g(x) = f(x - 1) + 2$ , what is the value of $g(3)$ ?