3.4 Inverse Functions and Transformations

Undoing a Function

The AEPA functions competency includes operations with functions, including composition and inverses. An inverse function reverses the input-output pairing of a one-to-one function. If f(4) = 9, then f^-1(9) = 4. This is not the same as 1/f(x). The notation f^-1 names an inverse function, not a reciprocal.

A function is one-to-one when different inputs always produce different outputs. Graphically, it passes the horizontal line test. Algebraically, f(a) = f(b) implies a = b. A function can fail the horizontal line test on its full domain but have an inverse on a restricted domain. For example, f(x) = x^2 is not one-to-one on all real numbers, but f(x) = x^2 with x >= 0 has inverse f^-1(x) = sqrt(x).

The domain and range swap. If f has domain [1, infinity) and range [4, infinity), then f^-1 has domain [4, infinity) and range [1, infinity). Graphs of inverse functions reflect across y = x, so points also swap: (a, b) on f becomes (b, a) on f^-1.

Algebraic Procedure

To find an inverse from an equation, use a consistent procedure:

1. Write y = f(x).
2. Interchange x and y.
3. Solve for y.
4. State the inverse domain and range using the original range and domain.
5. Verify by composition when needed.

Worked example: f(x) = sqrt(x - 1) + 4. The original domain is x >= 1 and the range is y >= 4. Write y = sqrt(x - 1) + 4. Swap variables: x = sqrt(y - 1) + 4. Then x - 4 = sqrt(y - 1). Squaring gives (x - 4)^2 = y - 1, so y = (x - 4)^2 + 1. The inverse is f^-1(x) = (x - 4)^2 + 1 with domain x >= 4 and range y >= 1. Without the inverse domain x >= 4, the equation alone hides the original range restriction.

Transformations: Before or After the Rule?

A transformation changes a parent graph in a predictable way. In g(x) = a f(b(x - h)) + k, the outside values a and k affect outputs. The graph shifts up k, down if k is negative, stretches vertically by |a|, and reflects across the x-axis if a is negative. The inside values b and h affect inputs. The graph shifts right h, left if h is negative, compresses horizontally by |b| if |b| > 1, and stretches horizontally if 0 < |b| < 1. A negative inside factor reflects across the y-axis.

Worked example: g(x) = -2 f((x + 3)/4) + 1. The +1 shifts outputs up 1. The -2 reflects across the x-axis and vertically stretches by 2. Inside, (x + 3)/4 means the graph is stretched horizontally by 4 and shifted left 3. The left shift is not 12 units; rewriting as (1/4)(x + 3) keeps the input structure clear.

Student Misconceptions

A common student claim is that f(x - 5) shifts left because of the minus sign. The misconception is reading the inside expression as an output operation. Since x must be 5 greater to produce the same original input, the graph shifts right 5. Another common error is to square both sides while finding an inverse and forget that squaring can require a domain restriction. The algebraic equation may be correct, but the inverse function must still respect the original range.

For teacher-certification reasoning, be ready to explain why a method works. The phrase switch x and y is a shortcut for reversing ordered pairs. The horizontal line test is a visual test for whether the reversal is still a function. Transformation rules come from asking what input would make the parent function see the same value as before.

Practice Strategy

When an item combines inverses and transformations, draw a quick map. Identify the parent function, state its domain and range, apply transformations in input-output order, and only then choose a graph or formula. If answer choices look similar, test one anchor point. For inverse choices, compose f(f^-1(x)) or f^-1(f(x)) with a simple value from the allowed domain. For transformation choices, track where a known point such as the vertex, endpoint, or intercept moves.

Domain, Range, and Anchor Points

Domain and range are the guardrails for both inverses and transformations. If a parent square-root graph begins at (0, 0), then y = 3sqrt(x - 2) - 5 begins at (2, -5), has domain x >= 2, and has range y >= -5. If that transformed function is inverted, the inverse domain begins at -5 and the inverse range begins at 2. Tracking the endpoint is often faster than deriving the full inverse formula.

Anchor points also expose incorrect transformation order. If f has point (1, 4), then g(x) = f(x - 3) + 2 has point (4, 6), because x = 4 makes the inside input 1 and the outside +2 changes the output. This concrete point-based reasoning is useful for explaining why horizontal shifts feel opposite without asking students to memorize a slogan.