Step-by-Step Solved Examples
Example 1: Set of Ordered Pairs
Find the inverse of \( f = \{( -2 , 0) , (0 , 1) , (2 , 3) , (3 , 4)\} \).
View Solution
To find the inverse of a set of points, simply swap the x and y coordinates:
\[ f^{-1} = \{(0 , -2) , (1 , 0) , (3 , 2) , (4 , 3)\} \]Domain of \( f^{-1} \): \(\{0, 1, 3, 4\}\)
Range of \( f^{-1} \): \(\{-2, 0, 2, 3\}\)
Example 2: Linear Function
Find the inverse of \( f(x) = 2x + 2 \).
View Solution
- Replace \( f(x) \) with \( y \): \( y = 2x + 2 \)
- Swap \( x \) and \( y \): \( x = 2y + 2 \)
- Solve for \( y \):
\( x - 2 = 2y \)
\( y = \frac{1}{2}x - 1 \)
Result: \( f^{-1}(x) = \frac{1}{2}x - 1 \)
Example 3: Cube Root Function
Find the inverse of \( g(x) = \sqrt[3]{x - 1} \).
View Solution
- Swap variables: \( x = \sqrt[3]{y - 1} \)
- Cube both sides: \( x^3 = y - 1 \)
- Solve for \( y \): \( y = x^3 + 1 \)
Result: \( g^{-1}(x) = x^3 + 1 \)
Example 4: Logarithmic Function
Find the inverse of \( h(x) = \ln(x - 1) \).
View Solution
- Swap variables: \( x = \ln(y - 1) \)
- Convert to exponential form: \( e^x = y - 1 \)
- Solve for \( y \): \( y = e^x + 1 \)
Result: \( h^{-1}(x) = e^x + 1 \)
Example 5: Square Root Function
Find the inverse of \( m(x) = \sqrt{x + 2} \).
View Solution
- Swap variables: \( x = \sqrt{y + 2} \)
- Square both sides: \( x^2 = y + 2 \)
- Solve for \( y \): \( y = x^2 - 2 \)
Critical Note: Since the range of the original function is \( y \ge 0 \), the domain of the inverse must be restricted.
Result: \( m^{-1}(x) = x^2 - 2 \text{ for } x \ge 0 \)
Example 6: Rational Function
Find the inverse of \( t(x) = \frac{1}{x-1} \).
View Solution
- Swap variables: \( x = \frac{1}{y-1} \)
- Multiply both sides by \((y-1)\): \( x(y-1) = 1 \)
- Expand and isolate \( y \):
\( xy - x = 1 \)
\( xy = 1 + x \)
\( y = \frac{x+1}{x} \)
Result: \( t^{-1}(x) = \frac{x+1}{x} \)
