How do you use inverse functions to solve logarithmic equations like log(x - 3) = 2?

Since logarithmic and exponential functions are inverses, rewrite log(x - 3) = 2 as x - 3 = 10^2, which yields x = 103.

How do you find the range of a function using its inverse?

The range of a one-to-one function is the domain of its inverse. By finding the inverse function, any domain restrictions determine the excluded values in the range of the original function.

Applications and Use of Inverse Functions

Examples on how to apply and use inverse functions in real-life situations and to solve problems in mathematics.

Example 1

Use inverse functions to solve equations.

Solve the following equation:

\[ \log(x - 3) = 2 \]

Solution to Example 1

Since logarithmic and exponential functions are inverses of each other, we can write: \[ A = \log(B) \quad \text{if and only if} \quad B = 10^A \]
Using this property, rewrite the given equation: \[ x - 3 = 10^2 \]
Solve for \(x\): \[ x = 103 \]

Example 2

Use inverse functions to find the range of a function.

Find the range of the function \(f\) given by:

\[ f(x) = \frac{2x}{x - 3} \]

Solution to Example 2

The range of a one-to-one function is the domain of its inverse. First, we show that \(f\) is one-to-one by starting with \(f(a) = f(b)\): \[ \frac{2a}{a - 3} = \frac{2b}{b - 3} \]
Multiply both sides by \((a-3)(b-3)\): \[ 2a(b - 3) = 2b(a - 3) \]
Expand and simplify: \[ 2ab - 6a = 2ab - 6b \implies -6a = -6b \implies a = b \]
Hence, the function is one-to-one. Now find its inverse by swapping \(x\) and \(y\): \[ x = \frac{2y}{y - 3} \]
Solve for \(y\): \[ x(y - 3) = 2y \implies xy - 3x = 2y \implies xy - 2y = 3x \implies y(x - 2) = 3x \] \[ f^{-1}(x) = \frac{3x}{x - 2} \]
The domain of \(f^{-1}\) excludes \(x = 2\). Hence, the range of \(f\) is all real numbers except 2.

Example 3

Use inverse functions to find the angle of elevation.

A camera photographs a hot-air balloon rising vertically. The camera is 300 meters from the launch point. The angle of elevation \(t\) depends on the height \(x\) of the balloon.

(a) Find \(t\) as a function of \(x\).
(b) Find \(t\) when \(x = 150, 300, 600\) meters.
(c) Graph \(t\) as a function of \(x\).

Solution to Example 3

Using right-triangle trigonometry: \[ \tan(t) = \frac{x}{300} \]
Apply the inverse tangent function: \[ t = \tan^{-1}\left(\frac{x}{300}\right) \]
Computed values of \(t\): \[ t(150) \approx 26.6^\circ, \quad t(300) = 45.0^\circ, \quad t(600) \approx 63.4^\circ \]

x	t (degrees)
0	0
150	26.6
300	45.0
600	63.4
1200	76.0
3000	84.3

Example 4

Use inverse functions to find the radius of a right circular cone.

Five cones of height \(h = 50\) cm have volumes 200, 400, 800, 1600, and 3200 cm\(^3\). Find the radius of each cone.

Solution to Example 4

Volume formula: \[ V = \frac{1}{3}\pi r^2 h \]
Solve for \(r\): \[ r = \sqrt{\frac{3V}{\pi h}} \]

\[ \begin{aligned} V = 200 &\Rightarrow r = 1.95\text{ cm} \\ V = 400 &\Rightarrow r = 2.76\text{ cm} \\ V = 800 &\Rightarrow r = 3.91\text{ cm} \\ V = 1600 &\Rightarrow r = 5.53\text{ cm} \\ V = 3200 &\Rightarrow r = 7.82\text{ cm} \end{aligned} \]

Example 5

Use inverse functions to solve population growth problems.

\[ P = 200{,}000 e^{0.01t} \]

Find when the population reaches 300,000; 400,000; and 500,000.

Solution to Example 5

Solve for \(t\) using the natural logarithm (inverse of exponential): \[ t = \frac{\ln(P/200{,}000)}{0.01} \]

\[ \begin{aligned} P = 300{,}000 &\Rightarrow t = 40.55 \text{ years (approx. 2041)} \\ P = 400{,}000 &\Rightarrow t = 69.31 \text{ years (approx. 2070)} \\ P = 500{,}000 &\Rightarrow t = 91.63 \text{ years (approx. 2092)} \end{aligned} \]