Relations & Functions
Inverse Functions
Grade 12
Question:
<p>If \(f(x) = ax + b\) and the equation \(f(x) = f^{-1}(x)\) is satisfied by every real value of \(x\), then which of the following is true?</p>
<p>(a) \(a = 2, b = -1\)</p>
<p>(b) \(a = -1, b \in \mathbb{R}\)</p>
<p>(c) \(a = 1, b \in \mathbb{R}\)</p>
<p>(d) \(a = 1, b = -1\)</p>
Step-by-Step Solution
Key Concept: For f(x) = f⁻¹(x) to hold for all x, the coefficients of x and the constant term must both be zero separately.
<p><strong>Solution:</strong></p><p>Given: $f(x) = ax + b$</p><p>The inverse is: $f^{-1}(x) = \frac{x - b}{a}$</p><p>Given condition: $f(x) = f^{-1}(x)$ for all $x \in \mathbb{R}$</p><p>$$ax + b = \frac{x - b}{a}$$</p><p>Multiply both sides by $a$:</p><p>$$a^2 x + ab = x - b$$</p><p>$$(a^2 - 1)x + ab + b = 0$$</p><p>For this to hold for all $x$:</p><p>$$a^2 - 1 = 0 \text{ and } ab + b = 0$$</p><p>$$a = \pm 1$$</p><p>If $a = 1$: $b + b = 0 \Rightarrow b = 0$ (but $b \in \mathbb{R}$ doesn't work universally)</p><p>If $a = -1$: $-b + b = 0$ (True for all $b \in \mathbb{R}$)</p><p>Hence, (b) is the correct answer.</p>
Correct Answer: b