Question

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is given by $f(x)=7x+8$ and $f^{-1}(12)=\dfrac{k}{7}$, then the value of $k$ is

• A. $7$
• B. $1$
• C. $4$
• D. $8$

Hint:

For linear functions, inverse functions are obtained by simple rearrangement.

Answer 1

The Correct Option is C

Step 1: Finding the inverse function.
\[ y = 7x + 8 \] Interchanging $x$ and $y$: \[ x = 7y + 8 \Rightarrow y = \frac{x-8}{7} \]
Step 2: Evaluating $f^{-1(12)$.}
\[ f^{-1}(12) = \frac{12-8}{7} = \frac{4}{7} \]
Step 3: Comparing with given value.
\[ \frac{k}{7} = \frac{4}{7} \Rightarrow k = 4 \]
Step 4: Conclusion.
The value of $k$ is $4$.