1.5 Inverse Functions and Logarithms - Quick Notes

Inverse Functions

• Definition: A function f is one-to-one if it never takes the same value twice; equivalently, $f(x<em>1)=f(x</em>2) \Rightarrow x<em>1=x</em>2.$ If f is one-to-one with domain A and range B, its inverse $f^{-1}$ has domain B and range A and satisfies $f^{-1}(f(x))=x\quad\text{and}\quad f(f^{-1}(y))=y$ for appropriate x,y.
• One-to-one and Horizontal Line Test:
  • A horizontal line intersects the graph of f at most once if and only if f is one-to-one.
• Existence of inverse:
  • Inverse exists iff f is one-to-one.
• Graphical interpretation:
  • The graph of $f^{-1}$ is the reflection of the graph of $f$ about the line $y=x$.
• How to compute inverse of a one-to-one function:
  1. Write $y=f(x)$.
  2. Solve the equation for x in terms of y.
  3. Swap x and y to obtain the inverse function $y=f^{-1}(x)$.
• Example:
  • If $f(x)=x^3$, then $f^{-1}(x)=\sqrt[3]{x}$, and $f^{-1}(f(x))=x,\; f(f^{-1}(x))=x$.
• Cancellation equations (inverse relationship):
  • $f^{-1}(f(x))=x\quad\text{and}\quad f(f^{-1}(y))=y$ show that f and $f^{-1}$ undo each other.

How to Find Inverse Functions (Steps)

• Step 1: Write $y=f(x)$.
• Step 2: Solve this equation for x in terms of y.
• Step 3: Interchange x and y to express the inverse as $f^{-1}(x)$.
• If f is not one-to-one, an inverse in the function sense may not exist.

Logarithmic Functions

• Exponential inverse:
  • For base $b>0, b\neq 1$, the exponential $f(x)=b^x$ is one-to-one and has inverse $f^{-1}=\log_b$ with domain $(0,\infty)$ and range $(-\infty,\infty)$.
• Definition:
  • $\log<em>b y = x \iff b^x = y.$ Equivalently, $f^{-1}(y)=\log</em>b y$.
• Graph and interpretation:
  • The graph of $\log_b$ is the reflection of $y=b^x$ about the line $y=x$; passes through (1,0); increasing for $b>1$.
• Domain and range:
  • $\log_b: (0,\infty)\to (-\infty,\infty)$.
• Change of base:
  • $\log_b x = \dfrac{\ln x}{\ln b}$.
• Laws of logarithms (for positive x,y):
  1. $\log<em>b(xy) = \log</em>b x + \log_b y$
  2. $\log<em>b(x^r) = r\log</em>b x$
  3. $\log<em>b\left(\dfrac{x}{y}\right) = \log</em>b x - \log_b y$
• Example:
  • $\log_8 5 = \dfrac{\ln 5}{\ln 8} \approx 0.773976$.
• Natural logarithms as a special case:
  • $\ln x = \log<em>e x$; Change of base gives $\log</em>b x = \dfrac{\ln x}{\ln b}$.
• Graph of $\ln x$:
  • Defined on $x>0$; increasing; vertical asymptote at $x=0$; grows slower than any positive power of x.
• Example: Sketch $y=\ln(x-2)-1$ by moving the graph of $\ln x$ right 2 and down 1.
• Growth remark:
  • $\ln x$ grows slower than any power $x^a$ with $a>0$.

Natural Logarithms

• Base e: $\ln x$ is the natural logarithm.
• Relationships:
  • $\ln x = \log_e x$; $e^{\ln x} = x$; $\ln(e^x) = x$; $\ln 1 = 0$.
• Example: Solve $\ln x = 5\Rightarrow x=e^5$.
• Change of base recap (special case): $\log_b x = \dfrac{\ln x}{\ln b}$.

Graph and Growth of the Natural Logarithm

• The graph of $y=\ln x$ is the reflection of the graph of $y=e^x$ about the line $y=x$.
• The natural logarithm is defined on $x>0$ and is increasing with a vertical asymptote at $x=0$.
• For large x, $\ln x$ grows much more slowly than polynomial functions of x.

Inverse Trigonometric Functions

• Key idea: Trigonometric functions are not one-to-one on their standard principal domains, so we restrict their domains to obtain inverses.
• Sine:
  • Restrict domain to $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$; inverse is $\arcsin$ with domain $[-1,1]$ and range $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$.
• Cosine:
  • Restrict domain to $[0, \pi]$; inverse is $\arccos$ with domain [-1,1] and range [0,\pi].
• Tangent:
  • Restrict domain to $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$; inverse is $\arctan$ with domain $\mathbb{R}$ and range $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$.
• Cancellation rules (with restricted domains):
  • $\sin(\arcsin x)=x,\quad \arcsin(\sin x)=x$ for x in the restricted range of arcsin.
  • Similar cancellation for cosine and tangent with their respective inverses.
• Graphical interpretation:
  • Inverse trig functions are reflections about the line $y=x$.
• Examples:
  • $\arcsin\left(\tfrac{1}{2}\right)=\tfrac{\pi}{6}$; $\arctan(\sqrt{3})=\tfrac{\pi}{3}$.
• Other inverse trig functions:
  • Inverses for cosecant, secant, and cotangent exist on appropriately restricted domains and are denoted by $\operatorname{arccsc}, \operatorname{arcsec}, \operatorname{arccot}$ with principal values.