Inverse Functions: Questions with Detailed Solutions

This page presents carefully selected questions on inverse functions with fully worked solutions and explanations. The goal is to strengthen both computational skills and conceptual understanding of inverse functions.

Questions and Solutions

Question 1

Find the parameters \(a\) and \(b\) of the linear function \[ f(x) = ax + b \] such that \[ f^{-1}(2) = 3 \quad \text{and} \quad f^{-1}(-3) = 6. \]

Solution

• From the definition of inverse functions: \[ f^{-1}(2) = 3 \Rightarrow f(3) = 2, \quad f^{-1}(-3) = 6 \Rightarrow f(6) = -3. \]
• Substitute into \(f(x) = ax + b\): \[ 3a + b = 2, \quad 6a + b = -3. \]
• Solving the system gives: \[ a = -\frac{5}{3}, \quad b = 7. \]

Question 2

Given that \( f(x) \) is an odd function and

Solution

• Since \(f(1) = 3\), it follows that: \[ f^{-1}(3) = 1. \]
• The function \(f\) is odd, so: \[ f(-2) = -f(2) = -12 \Rightarrow f^{-1}(-12) = -2. \]

Question 3

Prove that the inverse of an invertible odd function is also odd.

Solution

• By definition of inverse functions: \[ f(f^{-1}(x)) = x. \]
• Replace \(x\) with \(-x\): \[ f(f^{-1}(-x)) = -x. \]
• Since \(f\) is odd: \[ f(-u) = -f(u), \] which implies: \[ f(-f^{-1}(-x)) = x. \]
• Comparing both expressions: \[ f^{-1}(x) = -f^{-1}(-x), \] proving that \(f^{-1}\) is odd.

Question 4

Let \[ f(x) = \frac{1}{x - 2}. \] Find the points of intersection of the graphs of \(f\) and its inverse. Graph \(f\), \(f^{-1}\), and the line \(y = x\).

Solution

• Start with: \[ y = \frac{1}{x - 2}. \]
• Swap \(x\) and \(y\): \[ x = \frac{1}{y - 2}. \]
• Solve for \(y\): \[ y = \frac{1}{x} + 2 = f^{-1}(x). \]
• Solve: \[ \frac{1}{x - 2} = \frac{1}{x} + 2. \] This gives: \[ x = 1 \pm \sqrt{2}. \]
• The intersection points are: \[ (1 + \sqrt{2},\, 1 + \sqrt{2}), \quad (1 - \sqrt{2},\, 1 - \sqrt{2}). \]

Question 5

Graph the function \[ f(x) = |x - 2| + 2x, \] find its inverse, and graph both.

Solution

• For \(x < 2\): \[ f(x) = -(x - 2) + 2x = x + 2. \]
• For \(x \ge 2\): \[ f(x) = (x - 2) + 2x = 3x - 2. \]

• From the graph, sample points on \(f\) are: \[ (-2,0),\ (2,4),\ (3,7). \]
• Corresponding points on \(f^{-1}\) are: \[ (0,-2),\ (4,2),\ (7,3). \]
• Solving for the inverse gives: \[ f^{-1}(x) = -\frac{1}{3}|x - 4| + \frac{2}{3}x - \frac{2}{3}. \]

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