SAT Inverse Functions and Function Composition: Finding and Applying Inverses

The inverse of a function f, written f^-1, "undoes" what f does. If f(x)=2x+3, then f^-1(x)=(x-3)/2. You verify: f(f^-1(x))=f((x-3)/2)=2((x-3)/2)+3=x-3+3=x, confirming the inverse works. A function has an inverse only if it is one-to-one (injective), meaning each output corresponds to exactly one input. f(x)=x^2 is NOT one-to-one over all reals because f(2)=4 and f(-2)=4 (two inputs give the same output). To have an inverse, f(x)=x^2 must be restricted to x>=0 or x<=0; then it is one-to-one. A function is one-to-one if it passes the horizontal line test: any horizontal line crosses the graph at most once. If a horizontal line crosses the graph twice, the function is not one-to-one and does not have an inverse (unless restricted).

Domain and range swap for inverse functions. If f has domain [0,∞) and range [0,∞), then f^-1 has domain [0,∞) and range [0,∞). More generally, the domain of f^-1 equals the range of f, and the range of f^-1 equals the domain of f. Graphs of f and f^-1 are reflections across the line y=x; if the point (a,b) is on the graph of f, then (b,a) is on the graph of f^-1.

To find f^-1 algebraically, follow these steps: (1) Replace f(x) with y. (2) Swap x and y. (3) Solve for y. (4) Replace y with f^-1(x). Example: f(x)=2x+3. (1) y=2x+3. (2) x=2y+3. (3) x-3=2y, so y=(x-3)/2. (4) f^-1(x)=(x-3)/2. Another example: f(x)=x^2 (for x>=0). (1) y=x^2. (2) x=y^2. (3) y=sqrt(x) (taking the positive root since x>=0). (4) f^-1(x)=sqrt(x). When solving x=y^2 for y, take both roots (±sqrt(x)) unless the original function's domain restricts which one applies; paying attention to the domain prevents choosing the wrong inverse formula.

For more complex functions like f(x)=(2x+1)/(x-3), algebraic inversion involves solving (2y+1)/(y-3)=x for y. Multiply by (y-3): 2y+1=x(y-3)=xy-3x. Rearrange: 2y-xy=-3x-1, so y(2-x)=-3x-1, giving y=(-3x-1)/(2-x)=(3x+1)/(x-2). Then f^-1(x)=(3x+1)/(x-2). Verify: f(f^-1(x))=f((3x+1)/(x-2)). This algebraic verification is tedious but ensures correctness. On test day, check one value: if f^-1(7)=(3*7+1)/(7-2)=22/5, then f(22/5) should equal 7. Spot-checking one value is faster than full algebraic verification.

Function composition combines functions: (f∘g)(x)=f(g(x)) means apply g first, then apply f to the result. If f(x)=x+1 and g(x)=x^2, then (f∘g)(x)=f(g(x))=f(x^2)=x^2+1. Note that (f∘g)(x)≠(g∘f)(x) in general. (g∘f)(x)=g(f(x))=g(x+1)=(x+1)^2=x^2+2x+1. Composition order matters. For (f∘g)(5): g(5)=25, then f(25)=26, so (f∘g)(5)=26. For (g∘f)(5): f(5)=6, then g(6)=36, so (g∘f)(5)=36. To evaluate a composition, work from the inside out: evaluate the inner function first, then apply the outer function to that result.

Composition appears in questions asking you to evaluate expressions like f(g(2)) or to find (f∘g)(x) as a single function. The strategy is the same: identify which function is inner (applied first) and which is outer (applied second), then perform the operations in order. Errors occur when students confuse order or make arithmetic mistakes in intermediate steps. Check your work by evaluating at a test value, as shown above.

When asked to find an inverse, use the four-step algebraic method. When asked to evaluate an inverse (like finding f^-1(3)), substitute directly into your inverse formula or recognize that if f(a)=3, then f^-1(3)=a (use this shortcut if finding a is easier than finding the formula). When asked to verify an inverse, check that f(f^-1(x))=x or pick a specific value and verify f(f^-1(a))=a. A focused 2-week inverse drill: Day 1-3, find inverse functions for linear and quadratic (restricted) functions. Day 4-5, find inverses for rational functions. Day 6-7, mix all types. Days 8-14, practice composition, evaluation, and combinations of inverses and composition. By the end, you should find inverses mechanically and evaluate compositions quickly without hesitation.

Common mistakes to avoid: forgetting to restrict the domain for functions that need it (like f(x)=x^2), using the wrong inverse formula, and evaluating compositions in the wrong order. For each mistake, add a specific check to your routine: before finalizing an inverse, verify f(f^-1(x))=x at one value; when composing, explicitly label which function is applied first. These checks catch most errors.