1.4 Inverse Functions and Logarithms: Quick Notes

Inverse Functions

• One-to-one (injective): f(x1) = f(x2) implies x1 = x2.
• Horizontal Line Test: a function is one-to-one iff no horizontal line intersects its graph more than once.
• Existence of inverse: If f is one-to-one with domain A and range B, then the inverse f^{-1}: B → A is defined by f^{-1}(y) = x iff f(x) = y.
• Cancellation: $f^{-1}(f(x)) = x\quad\text{and}\quad f(f^{-1}(x)) = x.$
• How to find inverse: Solve y = f(x) for x in terms of y, then interchange x and y to obtain f^{-1}(x).
• Graphical view: The graph of f^{-1} is the reflection of the graph of f about the line $y = x.$
• Example: If $f(x) = x^3$, then $f^{-1}(x) = \sqrt[3]{x}$.
• Important note: Not all functions have an inverse (only one-to-one functions have inverses).

Logarithmic Functions

• If base $b>0, b\neq 1$, the exponential function $f(x)=b^x$ is one-to-one and has inverse $f^{-1}(x)=\log_b x.$
• Definition: $\log_b x$ is the exponent to which base $b$ must be raised to yield $x$; domain $x>0$; range $(-\infty, \infty)$.
• Cancellation: $\log<em>b(b^x)=x\quad\text{and}\quad b^{\log</em>b x}=x.$
• Change of base: $\log_b x = \dfrac{\ln x}{\ln b}.$
• Laws of logarithms (for positive $x,y$):
  • $\log<em>b(xy) = \log</em>b x + \log_b y$
  • $\log<em>b\left(\dfrac{x}{y}\right) = \log</em>b x - \log_b y$
  • $\log<em>b(x^r) = r\,\log</em>b x$
• Natural logarithm: base $e$, $\ln x = \log_e x$; inverses with exponential base $e$.
  • Properties: $\ln(e^x)=x\quad\text{and}\quad e^{\ln x}=x.$
  • Change of base example: $\log_8 5 = \dfrac{\ln 5}{\ln 8} \approx 0.773976.$
• Graphs: $y=\ln x$ is reflection of $y=e^x$ about $y=x$; domain $x>0$, vertical asymptote at $x=0$; increasing, grows slower than any power $x^a$.
• Special shifts: graphs like $\ln(x-2)-1$ obtained by horizontal/vertical shifts.

Natural Logarithms

• Natural log is base $e$: $\ln x = \log_e x$; inverse of $e^x$.
• Key facts: domain $x>0$; $\ln x = 5\Rightarrow x=e^5$.
• Change of base: same formula as above, with base change to compute logs of other bases.
• Graphical relation: the graph of $\ln x$ is the reflection of the graph of $e^x$ about the line $y=x$.

Graph and Growth of the Natural Logarithm

• $\ln x$ is increasing on $(0,\infty)$ with a vertical asymptote at $x=0$.
• It grows slower than any positive power of x: for large x, $\ln x \ll x^\alpha$ for any $\alpha>0$.

Inverse Trigonometric Functions

• Trigonometric functions are not one-to-one on their standard domains; restrict domains to obtain inverses:
  • $\sin x$ restricted to $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$ → inverse $\arcsin$ with domain $[-1,1]$ and range $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$.
  • $\cos x$ restricted to $[0, \pi]$ → inverse $\arccos$ with domain $[-1,1]$ and range $[0, \pi]$.
  • $\tan x$ restricted to $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$ → inverse $\arctan$ with domain $(-\infty, \infty)$ and range $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$.
• Inverses: arcsin, arccos, arctan; defined by the property $f^{-1}(f(x))=x$ and $f(f^{-1}(x))=x$ on their restricted domains.
• Graphical view: inverse trig functions are reflections about the line $y=x$.
• Examples: $\arcsin\left(\tfrac{1}{2}\right)=\tfrac{\pi}{6}$; $\arctan(3)$ is the angle in $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$ with tangent 3.
• Other inverses: the same idea applies to arcsec, arccsc, arccot with their respective principal values.