Relations & Functions
Inverse Functions
Grade 12
Question:
<p>If <math>f(x) = \frac{a - x}{a + x}</math>, the domain of <math>f^{-1}(x)</math> contains</p>
<p>(a) <math>(-7, 7)</math></p>
<p>(b) <math>(-7, -1)</math></p>
<p>(c) <math>(-1, 7)</math></p>
<p>(d) <math>(0, 7)</math></p>
Step-by-Step Solution
Key Concept: The domain of the inverse function is the range of the original function. Find where the inverse is undefined.
<p>Let <math>y = f(x) = \frac{a - x}{a + x}</math>. Then <math>ay + xy = a - x</math>.</p><p>Solving for <math>x</math>: <math>x = \frac{a(1-y)}{1+y} = f^{-1}(y)</math>.</p><p>Thus <math>f^{-1}(x) = \frac{a(1-x)}{1+x}</math>.</p><p><math>f^{-1}(x)</math> is not defined when <math>x = -1</math>.</p><p>Domain of <math>f^{-1}(x)</math> is <math>(-\infty, -1) \cup (-1, \infty)</math>.</p><p>When <math>a = -1</math>, <math>f(x) = -1</math>, which is constant. Then <math>f^{-1}(x)</math> is not defined.</p><p>Therefore, the domain of <math>f^{-1}(x)</math> belongs to <math>(-\infty, -1) \cup (-1, \infty)</math>, which contains <math>(-7, -1)\cup(-1, 7)\cup(0, 7)</math>.</p>
Correct Answer: b, c, d