Essential Concepts
Composition of Functions
Function composition is combining two functions so that the output of one function becomes the input of another. Think of it like a two-step process: the first function processes the input, then the second function processes that result.
The composition of functions [latex]f[/latex] and [latex]g[/latex] is written as [latex](f \circ g)(x) = f(g(x))[/latex], read as “f of g of x” or “f composed with g at x.”
In general, [latex](f \circ g)(x) \neq (g \circ f)(x)[/latex]. The order in which you compose functions matters because you’re applying different operations in different sequences.
Evaluating Composite Functions
- From formulas: Substitute the inner function into the outer function
- From tables: Find the output of the inner function first, then use that as the input for the outer function
- From graphs: Read the output of the inner function from its graph, then use that value as the input on the outer function’s graph
Domain of Composite Functions The domain of [latex](f \circ g)(x)[/latex] consists of all [latex]x[/latex] values that satisfy both:
- [latex]x[/latex] must be in the domain of [latex]g[/latex]
- [latex]g(x)[/latex] must be in the domain of [latex]f[/latex]
Decomposing Functions Sometimes you need to break down a complex function into simpler parts. Look for a “function inside a function” pattern. There may be multiple ways to decompose the same function.
Transformation of Functions
Types of Transformations
Vertical Shifts: [latex]g(x) = f(x) + k[/latex]
- Add [latex]k[/latex] to shift up (when [latex]k > 0[/latex])
- Subtract [latex]k[/latex] to shift down (when [latex]k < 0[/latex])
- Affects the output values
Horizontal Shifts: [latex]g(x) = f(x - h)[/latex]
- Subtract [latex]h[/latex] to shift right (when [latex]h > 0[/latex])
- Add [latex]h[/latex] to shift left (when [latex]h < 0[/latex])
- Affects the input values
Vertical Reflections: [latex]g(x) = -f(x)[/latex]
- Multiply the output by [latex]-1[/latex]
- Reflects the graph across the x-axis
Horizontal Reflections: [latex]g(x) = f(-x)[/latex]
- Multiply the input by [latex]-1[/latex]
- Reflects the graph across the y-axis
Vertical Stretches and Compressions: [latex]g(x) = a \cdot f(x)[/latex]
- If [latex]|a| > 1[/latex]: vertical stretch (graph becomes taller)
- If [latex]0 < |a| < 1[/latex]: vertical compression (graph becomes shorter)
- Affects the output values
Horizontal Stretches and Compressions: [latex]g(x) = f(bx)[/latex]
- If [latex]|b| > 1[/latex]: horizontal compression (graph becomes narrower) by factor [latex]\frac{1}{b}[/latex]
- If [latex]0 < |b| < 1[/latex]: horizontal stretch (graph becomes wider) by factor [latex]\frac{1}{b}[/latex]
- Affects the input values
Even and Odd Functions
- Even functions: Symmetric about the y-axis, satisfy [latex]f(x) = f(-x)[/latex]
- Odd functions: Symmetric about the origin, satisfy [latex]f(x) = -f(-x)[/latex]
- A function can be even, odd, or neither
Order of Transformations When applying multiple transformations to [latex]y = a \cdot f(b(x - h)) + k[/latex], apply in this order:
- Horizontal shifts ([latex]h[/latex])
- Horizontal stretches/compressions ([latex]b[/latex])
- Reflections (negative signs on [latex]a[/latex] or [latex]b[/latex])
- Vertical stretches/compressions ([latex]a[/latex])
- Vertical shifts ([latex]k[/latex])
Inverse Functions
An inverse function reverses the operation of the original function. If [latex]f[/latex] takes input [latex]a[/latex] to output [latex]b[/latex], then [latex]f^{-1}[/latex] takes input [latex]b[/latex] back to output [latex]a[/latex].
The inverse of [latex]f(x)[/latex] is written [latex]f^{-1}(x)[/latex], read as “f inverse of x.” Note: This is NOT the same as [latex]\frac{1}{f(x)}[/latex].
A function must be one-to-one (pass the horizontal line test) to have an inverse. If a function is not one-to-one over its entire domain, you may be able to restrict the domain to make it one-to-one.
Two functions [latex]f[/latex] and [latex]g[/latex] are inverses if both of these are true:
- [latex]f(g(x)) = x[/latex] for all [latex]x[/latex] in the domain of [latex]g[/latex]
- [latex]g(f(x)) = x[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]
Domain and Range of Inverses
- The domain of [latex]f[/latex] = the range of [latex]f^{-1}[/latex]
- The range of [latex]f[/latex] = the domain of [latex]f^{-1}[/latex]
Finding Inverse Functions
- Verify the function is one-to-one
- Replace [latex]f(x)[/latex] with [latex]y[/latex]
- Interchange [latex]x[/latex] and [latex]y[/latex]
- Solve for [latex]y[/latex]
- Replace [latex]y[/latex] with [latex]f^{-1}(x)[/latex]
Graphing Inverse Functions The graph of [latex]f^{-1}(x)[/latex] is the reflection of the graph of [latex]f(x)[/latex] across the line [latex]y = x[/latex]. If point [latex](a, b)[/latex] is on the graph of [latex]f[/latex], then point [latex](b, a)[/latex] is on the graph of [latex]f^{-1}[/latex].
Glossary
composite function
the new function formed by function composition, when the output of one function is used as the input of another
even function
a function whose graph is unchanged by horizontal reflection, satisfying [latex]f(x) = f(-x)[/latex], and is symmetric about the y-axis
horizontal compression
a transformation that compresses a function’s graph horizontally by multiplying the input by a constant [latex]b > 1[/latex]
horizontal reflection
a transformation that reflects a function’s graph across the y-axis by multiplying the input by [latex]-1[/latex]
horizontal shift
a transformation that shifts a function’s graph left or right by adding a constant to the input
horizontal stretch
a transformation that stretches a function’s graph horizontally by multiplying the input by a constant [latex]0 < b < 1[/latex]
inverse function
for any one-to-one function [latex]f(x)[/latex], the inverse is a function [latex]f^{-1}(x)[/latex] such that [latex]f^{-1}(f(x)) = x[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]; this also implies that [latex]f(f^{-1}(x)) = x[/latex] for all [latex]x[/latex] in the domain of [latex]f^{-1}[/latex]
odd function
a function whose graph is unchanged by combined horizontal and vertical reflection, satisfying [latex]f(x) = -f(-x)[/latex], and is symmetric about the origin
one-to-one function
a function where each output value corresponds to exactly one input value; passes the horizontal line test
vertical compression
a transformation that compresses a function’s graph vertically by multiplying the output by a constant [latex]0 < a < 1[/latex]
vertical reflection
a transformation that reflects a function’s graph across the x-axis by multiplying the output by [latex]-1[/latex]
vertical shift
a transformation that shifts a function’s graph up or down by adding a constant to the output
vertical stretch
a transformation that stretches a function’s graph vertically by multiplying the output by a constant [latex]a > 1[/latex]