3.3 Inverse Functions and Function Transformations
Key Takeaways
- A function f possesses an inverse f^(-1) if and only if it is one-to-one, passing the Horizontal Line Test on its graph.
- The domain of f^(-1) is equal to the range of f, and the range of f^(-1) is equal to the domain of f, with their graphs reflecting symmetrically across the line y = x.
- Transformations modify parent functions: f(x ± c) shifts horizontally, f(x) ± c shifts vertically, -f(x) reflects over the x-axis, and f(-x) reflects over the y-axis.
- Non-rigid transformations scale graphs: a * f(x) stretches (|a|>1) or compresses (0<|a|<1) vertically, while f(b * x) compresses (|b|>1) or stretches (0<|b|<1) horizontally.
3.3 Inverse Functions and Function Transformations
1. One-to-One Functions and Invertibility
A function $f$ is one-to-one (injective) if no two distinct inputs in the domain produce the same output value in the range. Formally:
The Horizontal Line Test (HLT)
Graphically, a function $f$ has an inverse function $f^{-1}$ if and only if no horizontal line intersects the graph of $f$ at more than one point.
- If a horizontal line touches the graph multiple times, the function is not one-to-one, meaning multiple inputs map to the same output.
One-to-One Function (Passes HLT): Not One-to-One (Fails HLT):
y ^ y ^ ----- (2 intersections)
| / | / \
| / | / \
| / |/_____\____
----+-------> x ----+-------------> x
2. Concept and Algebra of Inverse Functions
If $f$ is a one-to-one function with domain $A$ and range $B$, then its inverse function $f^{-1}$ has domain $B$ and range $A$, defined by:
Fundamental Properties of Inverse Functions
- Cancellation Equations:
- Domain-Range Swap:
- Line of Symmetry: The graphs of $y = f(x)$ and $y = f^{-1}(x)$ are symmetric reflections of each other across the identity line $y = x$.
Step-by-Step Procedure to Find $f^{-1}(x)$ Algebraically
- Verify $f(x)$ is one-to-one (or restrict its domain if necessary).
- Replace $f(x)$ with $y$.
- Swap the variables $x$ and $y$ ($x \to y$ and $y \to x$).
- Solve the resulting equation for $y$ in terms of $x$.
- Replace $y$ with notation $f^{-1}(x)$ and state any domain restrictions.
Worked Example 2.3A: Finding the Inverse of a Rational Function
Find the inverse of $f(x) = \frac{4x + 1}{2x - 3}$ for $x \neq \frac{3}{2}$.
Step 1: Replace $f(x)$ with $y$ and swap $x$ and $y$.
Step 2: Clear fraction and expand.
Step 3: Collect $y$ terms on one side.
Step 4: Solve for $y$.
Step 5: Write final inverse notation. (Notice: Domain of $f^{-1}$ is $(-\infty, 2) \cup (2, \infty)$, which matches the range of $f$.)
3. Function Transformations
Starting from basic parent functions—such as $f(x) = x^2, \sqrt{x}, |x|, \frac{1}{x}, x^3$—we can build complex function models by applying systematic transformations.
Transformation Rules Summary Table
| Transformation Type | Function Algebra | Effect on Graph | Coordinate Effect $(x,y)$ |
|---|---|---|---|
| Vertical Shift Up | $g(x) = f(x) + k \quad (k>0)$ | Shift graph up by $k$ units | $(x, y) \to (x, y+k)$ |
| Vertical Shift Down | $g(x) = f(x) - k \quad (k>0)$ | Shift graph down by $k$ units | $(x, y) \to (x, y-k)$ |
| Horizontal Shift Right | $g(x) = f(x - h) \quad (h>0)$ | Shift graph right by $h$ units | $(x, y) \to (x+h, y)$ |
| Horizontal Shift Left | $g(x) = f(x + h) \quad (h>0)$ | Shift graph left by $h$ units | $(x, y) \to (x-h, y)$ |
| Vertical Reflection | $g(x) = -f(x)$ | Reflect across $x$-axis | $(x, y) \to (x, -y)$ |
| Horizontal Reflection | $g(x) = f(-x)$ | Reflect across $y$-axis | $(x, y) \to (-x, y)$ |
| Vertical Stretch | $g(x) = a \cdot f(x) \quad ( | a | >1)$ |
| Vertical Compression | $g(x) = a \cdot f(x) \quad (0< | a | <1)$ |
| Horizontal Compression | $g(x) = f(b \cdot x) \quad ( | b | >1)$ |
| Horizontal Stretch | $g(x) = f(b \cdot x) \quad (0< | b | <1)$ |
4. Standard Form and Order of Transformations
When multiple transformations occur simultaneously, their order matters. We express a general transformed function in standard factored form:
Proper Sequence of Transformation Steps
- Horizontal Shifts: Shift left or right by $h$.
- Horizontal Stretching/Compressing & Reflection: Scale horizontally by factor $\frac{1}{|b|}$; reflect across $y$-axis if $b < 0$.
- Vertical Stretching/Compressing & Reflection: Scale vertically by factor $|a|$; reflect across $x$-axis if $a < 0$.
- Vertical Shifts: Shift up or down by $k$.
Worked Example 2.3B: Deconstructing Multi-Step Transformations
Describe the sequence of transformations that converts $f(x) = \sqrt{x}$ into $g(x) = -3\sqrt{2x - 8} + 5$.
Step 1: Factor the argument inside the square root into standard form.
Step 2: Identify transformations in sequential order.
- Horizontal shift right by $4$ units (due to $(x - 4)$).
- Horizontal compression by a factor of $\frac{1}{2}$ (due to $b = 2$).
- Vertical stretch by a factor of $3$ (due to $|a| = 3$).
- Vertical reflection across the $x$-axis (due to $a = -3 < 0$).
- Vertical shift up by $5$ units (due to $+5$).
5. CLEP Precalculus Exam Traps
| Common Trap | Incorrect Assumption | Correct Mathematical Rule |
|---|---|---|
| Unfactored Horizontal Shift | Reading $f(2x - 6)$ as a horizontal shift right of 6 units | Must factor first: $f(2(x - 3))$, which reveals a horizontal shift right of 3 units. |
| Inverse Notation Error | Confusing $f^{-1}(x)$ with exponent $\frac{1}{f(x)}$ | $f^{-1}(x)$ denotes the inverse function, NOT the reciprocal $\frac{1}{f(x)} = (f(x))^{-1}$. |
| Failing Horizontal Line Test | Attempting to find an inverse for $f(x) = x^2$ on domain $(-\infty, \infty)$ | Non-one-to-one functions require a restricted domain (e.g., $x \ge 0$) to possess an inverse. |
What is the inverse function f^{-1}(x) for f(x) = (2x - 7) / (x + 3) where x \neq -3?
The graph of g(x) = -3(x + 4)^2 - 5 is obtained from parent function f(x) = x^2 by which sequence of transformations?
If the domain of a one-to-one function f(x) = \sqrt{x - 3} + 2 is [3, \infty), what is the domain of its inverse function f^{-1}(x)?
How does the graph of h(x) = f(2x) compare to the graph of the parent function f(x)?