Lecture 9: Operations on Functions (continued) — Inverse Functions & Implicit vs. Explicit

1. Inverse Functions

1.1 Definition & Existence

An inverse function $f^{-1}$ “undoes” $f$:

$$f^{-1}(f(x)) = x,\quad f(f^{-1}(x)) = x.$$

A function $f$ has an inverse if and only if it is one-to-one (injective) and onto its range.

1.2 One-to-One & the Horizontal Line Test

• One-to-One: each $y$ in the range is hit by exactly one $x$.
• Horizontal Line Test: no horizontal line intersects the graph of $f$ more than once.

Example:

• $f(x)=x^3$ passes the test → invertible on $\mathbb{R}$.
• $g(x)=x^2$ fails on $\mathbb{R}$ (two $x$’s for positive $y$) → not invertible unless domain restricted.

2. Finding an Inverse Algebraically

1. Write $y = f(x)$.
2. Swap $x$ and $y$: $x = f(y)$.
3. Solve for $y$; that gives $f^{-1}(x)$.
4. State domain of $f^{-1}$ = range of $f$, and vice versa.

Example:
$f(x)=\dfrac{x-1}{x+2}$.

$$y = \frac{x-1}{x+2} \;\Longrightarrow\; x = \frac{y-1}{y+2} \;\Longrightarrow\; x(y+2)=y-1 \;\Longrightarrow\; y(x-1) = -2x-1$$

$$\Longrightarrow \boxed{f^{-1}(x) = \frac{-2x - 1}{\,x - 1\,}.}$$

3. Graphing Inverses

• Plot $y = f(x)$ and $y = f^{-1}(x)$ together.
• Draw the line $y = x$.
• Observe the mirror symmetry: each point $(a,b)$ on $f$ corresponds to $(b,a)$ on $f^{-1}$.

Tip: Reflect landmark points (intercepts, extrema) across the line $y=x$.

4. Implicit vs. Explicit Functions

4.1 Explicit Form

$$y = f(x).$$

– Direct formula for $y$.
– Easy to evaluate and plot.

4.2 Implicit Form

$$F(x,y) = 0,$$

where solving for $y$ may require multiple branches or be impossible in closed form (i.e. no single $y=$ expression in $x$).

Example (Circle):

$$x^2 + y^2 - 9 = 0.$$

Explicit branches:
$\displaystyle y = +\sqrt{9 - x^2},\quad y = -\sqrt{9 - x^2}.$

5. When to Use Implicit Representation

• Vertical tangents or cusps (hard to express $y$ as a single-valued function).
• Closed loops or self-intersecting curves.
• Relations that are not global functions but split into branches.

Example (Folium):
$\displaystyle x^3 + y^3 - 3xy = 0$
– Has a loop; no single explicit $y=f(x)$ covering all points.

Regular Exercises

1. Inverse Practice
   a) $f(x)=3x-5$
   b) $f(x)=\sqrt{x+2}$
   c) $f(x)=\dfrac{2x+1}{x-1}$

2. Horizontal Line Test
   Determine if each is invertible on $\mathbb{R}$:
   a) $y = x^2 - 4$
   b) $y = 2^x$
   c) $y = \log_{10} x$

3. Graph & Reflect

   • Plot $f(x)=x^3 - 3x$.
   • On $[1,\infty)$, find and plot its inverse branch.
   • Include the line $y = x$.

4. Inverse Composition
   Let $f(x)=2x+3$, $g(x)=\sqrt{x}$.

   • Compute $(f\circ g)^{-1}$.
   • Compare with $g^{-1}\circ f^{-1}$.

5. Implicit → Explicit
   Given $x^2 + xy + y^2 - 7 = 0$:

   • Solve for $y$ to find both branches.
   • State the domain for each branch.

6. Implicit Curve Analysis
   For $x^4 + y^4 = 8xy$:

   • Sketch or graph the relation.
   • Identify loops or intersections.
   • Explain why no single explicit $y=f(x)$ exists.