10.1 Finding Composite and Inverse Functions

Composite Functions

Construction of composite functions

A composite function is what you get when you feed the output of one function directly into another. The notation $(f \circ g)(x)$ means "apply $g$ first, then apply $f$ to the result." This order trips people up constantly, so pay close attention: the function closest to $x$ goes first.

To build a composite function algebraically:

1. Start with the inner function $g(x)$ and the outer function $f(x)$.
2. Wherever you see $x$ in the formula for $f$, replace it with the entire expression for $g(x)$.
3. Simplify the result.

Example 1: If $f(x) = x^2$ and $g(x) = x + 1$, then:

$(f \circ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1$

Example 2: If $f(x) = 2x - 1$ and $g(x) = 3x + 2$, then:

$(f \circ g)(x) = f(g(x)) = 2(3x + 2) - 1 = 6x + 4 - 1 = 6x + 3$

Notice that $(f \circ g)(x)$ and $(g \circ f)(x)$ are usually not the same thing. Using Example 2, $(g \circ f)(x) = g(f(x)) = 3(2x - 1) + 2 = 6x - 1$. Different result. Order matters.

You can also evaluate composites graphically by tracing values through two graphs:

1. Start at $x$ on the horizontal axis and read up to the graph of $g$ to find $g(x)$.
2. Take that $g(x)$ value and use it as the input on the horizontal axis for $f$.
3. Read up to the graph of $f$ to find $f(g(x))$.

This is useful for reading composite values off a graph when you don't have formulas.

Inverse Functions

Identification of one-to-one functions

Before a function can have an inverse, it must be one-to-one: every output comes from exactly one input. If two different inputs ever produce the same output, the function isn't one-to-one, and you can't reverse it unambiguously.

The quick visual check is the horizontal line test. Draw (or imagine) horizontal lines across the graph. If every horizontal line hits the graph at most once, the function is one-to-one.

• $f(x) = x^3$ passes the horizontal line test, so it's one-to-one.
• $f(x) = x^2$ fails because, for example, both $x = 3$ and $x = -3$ give the output 9.

Only one-to-one functions have inverses. When a function isn't one-to-one, you can sometimes restrict its domain to make it one-to-one (this is exactly what's done with $f(x) = x^2$ when we restrict to $x \geq 0$).

Determination of inverse functions

An inverse function $f^{-1}(x)$ reverses what the original function does. If $f(a) = b$, then $f^{-1}(b) = a$. The notation $f^{-1}$ does not mean $\frac{1}{f}$; it means the function that undoes $f$.

Algebraic method for finding an inverse:

1. Replace $f(x)$ with $y$: write $y = f(x)$.
2. Swap $x$ and $y$ in the equation.
3. Solve the new equation for $y$.
4. Replace $y$ with $f^{-1}(x)$.

Example: Find the inverse of $f(x) = 2x + 1$.

1. $y = 2x + 1$
2. $x = 2y + 1$
3. $x - 1 = 2y$, so $y = \frac{x - 1}{2}$
4. $f^{-1}(x) = \frac{x - 1}{2}$

You can verify by checking that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. This composition check is the definitive test for whether two functions are truly inverses.

Graphical method: The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$. Every point $(a, b)$ on the graph of $f$ becomes $(b, a)$ on the graph of $f^{-1}$.

Domain and range of inverse functions

The domain and range swap between a function and its inverse:

• The domain of $f^{-1}$ equals the range of $f$.
• The range of $f^{-1}$ equals the domain of $f$.

Example: $f(x) = \sqrt{x}$ has domain $x \geq 0$ and range $y \geq 0$. Its inverse is $f^{-1}(x) = x^2$, but only for $x \geq 0$ (the domain of $f^{-1}$ is restricted to match the range of $f$). Without that restriction, $x^2$ isn't one-to-one and wouldn't truly be an inverse.

Function Notation and Transformations

Function notation like $f(x) = \text{expression}$ gives you a compact way to name a function and communicate exactly what it does to an input. When you see $f(3)$, it means "plug 3 in for $x$."

This notation also makes it easy to describe transformations of graphs:

• Vertical shift: $f(x) + k$ shifts the graph up by $k$ units (down if $k$ is negative).
• Horizontal shift: $f(x - h)$ shifts the graph right by $h$ units (left if $h$ is negative).
• Vertical stretch/compression: $a \cdot f(x)$ stretches the graph vertically by a factor of $a$ (compresses if $0 < a < 1$).
• Reflection: $-f(x)$ reflects across the $x$-axis; $f(-x)$ reflects across the $y$-axis.

These transformations connect directly to composites and inverses. For instance, reflecting $f(x)$ across the line $y = x$ is exactly how you graph $f^{-1}(x)$.