Inverse of the function f(x)=10^x-10^-x10^x+10^-x is

B. 1/2log_10(1+x/1-x). Step 1: Put y=10^x-10^-x10^x+10^-x and multiply top and bottom by 10^x: y=10^2x-110^2x+1. Step 2: Cross-multiply: y⋅10^2x+y=10^2x-1, so 10^2x=1+y/1-y. Step 3: Take log_10: x=1/2log_10(1+y/1-y). Renaming y as x gives option (B). Check: f(0)=0, and option (B) at x=0 gives 1/2log_101=0. Trap: (A) log_10(2-x) at x=0 gives log_102≠ 0, but f^-1(0) must be 0.

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NIMCET 2022

Inverse of the function $f (x) = \frac{10 ^{x} - 10 ^{- x}}{10 ^{x} + 10 ^{- x}}$ is

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — $\frac{1}{2} l o g_{10} (\frac{1 + x}{1 - x})$

Detailed Solution

Put $y = \frac{10 ^{x} - 10 ^{- x}}{10 ^{x} + 10 ^{- x}}$ and multiply top and bottom by $10^{x}$ : $y = \frac{10 ^{2 x} - 1}{10 ^{2 x} + 1}$ .

Cross-multiply: $y \cdot 10^{2 x} + y = 10^{2 x} - 1$ , so $10^{2 x} = \frac{1 + y}{1 - y}$ .

Take $l o g_{10}$ : $x = \frac{1}{2} l o g_{10} (\frac{1 + y}{1 - y})$ . Renaming $y$ as $x$ gives option (B).

✓ Check

f (0) = 0

, and option (B) at

x = 0

gives

\frac{1}{2} l o g_{10} 1 = 0

⚠ Common trap

(A)

l o g_{10} (2 - x)

x = 0

gives

l o g_{10} 2 \neq = 0

, but

f^{- 1} (0)

must be

0

Quick tip

$y = \frac{t - 1}{t + 1} \Rightarrow t = \frac{1 + y}{1 - y}$ , so $f^{- 1} (x) = \frac{1}{2} lo g_{10} \frac{1 + x}{1 - x}$

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Functions — Preparation

This question tests your understanding of Functions, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A function assigns exactly one output to each input in its domain. This topic covers the domain, codomain and range of a function, types of functions (one-one, onto, even and odd functions), and standard functions such as the identity, constant, polynomial, rational, modulus, signum, greatest integer, exponential and logarithmic functions, along with their graphs. It also covers the composition of functions, the inverse of a function, and finding the domain and range of a given expression. For the domain, look for values that make a denominator zero, the expression under a square root negative, or the argument of a logarithm non-positive, and exclude them. A function has an inverse only if it is both one-one and onto. Quick graph sketches help a great deal with questions on range and on the number of solutions of an equation. Practise the functions questions below, each with a step-by-step solution.

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