1.6

Inverse Functions and Logarithms

Functions: Definitions, Domain, and Range

A function \( f(x) \) is a rule that assigns to each element \( x \) in a set \( D \) (the domain) exactly one element \( f(x) \) in a set \( R \) (the range).

• Polynomials: Domain is \(( -\infty, \infty )\).

• Radicals: Domain is restricted so the expression under the root is non-negative.

• Rational Functions: Domain excludes values that make the denominator zero.

One-to-One and Not One-to-One Functions

A function is one-to-one if every horizontal line intersects its graph at most once. Otherwise, it is not one-to-one.

• One-to-one functions have inverses that are also functions.

• Not one-to-one functions do not have inverses that are functions unless their domains are restricted.

Inverse Functions: Definition and Properties

If \( f \) is a one-to-one function on domain \( D \) with range \( R \), the inverse function \( f^{-1} \) is defined by:

• \( f^{-1}(b) = a \) if \( f(a) = b \).

• The domain of \( f^{-1} \) is \( R \), and the range of \( f^{-1} \) is \( D \).

Finding the Inverse of a Function Algebraically

The process of finding the inverse of a function \( f(x) \) involves:

1. Replace \( f(x) \) with \( y \).

2. Interchange \( x \) and \( y \).

3. Solve the equation for \( y \).

4. Replace \( y \) with \( f^{-1}(x) \).

Example: Find the inverse of \( f(x) = 5x - 7 \):

• Let \( y = 5x - 7 \).

• Interchange: \( x = 5y - 7 \).

• Solve for \( y \): \( y = \frac{x + 7}{5} \).

• So, \( f^{-1}(x) = \frac{x + 7}{5} \).

Graphs of Inverse Functions

The graph of \( y = f^{-1}(x) \) is obtained by reflecting the graph of \( y = f(x) \) about the line \( y = x \). The domain and range are interchanged between a function and its inverse.

Exponential and Logarithmic Functions

The natural exponential function is \( f(x) = a^x \) where \( a > 0 \). Its inverse is the logarithmic function, denoted \( f(x) = \log_a x \), where \( a > 0, a \neq 1 \).

• Domain of exponential: \( ( -\infty, \infty ) \)

• Range of exponential: \( (0, \infty) \)

• Domain of logarithm: \( (0, \infty) \)

• Range of logarithm: \( ( -\infty, \infty ) \)

The natural logarithmic function is \( f(x) = \ln x \), the inverse of \( f(x) = e^x \).

Properties of Logarithms

Logarithms have several important algebraic properties that allow us to expand or condense expressions:

Rule	Formula
Product Rule	\( \ln(bx) = \ln b + \ln x \)
Quotient Rule	\( \ln\left(\frac{b}{x}\right) = \ln b - \ln x \)
Reciprocal Rule	\( \ln\left(\frac{1}{x}\right) = -\ln x \)
Power Rule	\( \ln(x^r) = r \ln x \)

Inverse Trigonometric Functions

The six trigonometric functions are not one-to-one over their entire domains due to periodicity. By restricting their domains, we can define their inverses:

• \( \sin^{-1} x = y \) means \( \sin y = x \), also written as \( y = \arcsin x \).

• Similar definitions apply for \( \arccos x \) and \( \arctan x \).

Example:

• \( \cos^{-1}(-1) = \pi \)

• \( \arctan(-1) = -\frac{\pi}{4} \)

• \( \arcsin(\sin \pi) = 0 \)

Summary Table: Domains and Ranges of Key Functions

Function	Domain	Range
Polynomial	\( ( -\infty, \infty ) \)	\( ( -\infty, \infty ) \)
Radical (e.g., \( \sqrt{x} \))	\( [0, \infty) \)	\( [0, \infty) \)
Rational (e.g., \( \frac{x+1}{x-5} \))	\( x \neq 5 \)	Depends on function
Exponential (\( a^x \))	\( ( -\infty, \infty ) \)	\( (0, \infty) \)
Logarithmic (\( \log_a x \))	\( (0, \infty) \)	\( ( -\infty, \infty ) \)