Question

There are three function given in column-A and its inverse in column-B \begin{tabular}{l|l} \(\frac{\text { Column-A }}{\text { (1) } f(x)=1-2^{-x}}\) & (a) \(\left.f^{-1}(x)=\left[x / \sqrt{(}-x^{2}\right)\right]\) \\ (2) \(f(x)=\sin \left(\tan ^{-1} x\right)\) & (b) \(f^{-1}(x)=-\log _{2}(1-x)\) \\\ (3) \(f(x)=2 x+3\) & (c) \(f^{-1}(x)=[(x-3) / 2]\) \end{tabular} which one of the following matches is correct? (a) (1) \(\mathrm{a},(2) \mathrm{b},(3) \mathrm{c}\) (b) (1) b, (2) c, (3) a (c) (1) b, (2) a, (3) c (d) (1) c, (2) b, (3) a

Short answer

The correct match is: (a) (1) c, (2) b, (3) a.

Step-by-step solution

Testing Option (a)

Option (a) states (1) a, (2) b, (3) c. Let's test it. For (1): \(f(x)=1-2^{-x}\) and \(f^{-1}(x)=\frac{x}{\sqrt{-x^2}}\) Substitute \(f^{-1}(x)\) in \(f(x)\): \[f(f^{-1}(x)) = 1-2^{-\frac{x}{\sqrt{-x^2}}}\] Since f(f^{-1}(x)) ≠ x, option (a) is not correct.

Testing Option (b)

Option (b) states (1) b, (2) c, (3) a. Let's test it. For (1): \(f(x)=1-2^{-x}\) and \(f^{-1}(x)=-\log_2(1-x)\) Substitute \(f^{-1}(x)\) in \(f(x)\): \[f(f^{-1}(x)) = 1-2^{-(-\log_2(1-x))}\] \[f(f^{-1}(x)) = 1-(1-x)\] \[f(f^{-1}(x)) = x\] For (2): \(f(x)=\sin(\tan^{-1}(x))\) and \(f^{-1}(x)=\frac{(x-3)}{2}\) It can be seen that the function f(x) and its inverse are not the same form. Thus, option (b) is not correct.

Testing Option (c)

Option (c) states (1) b, (2) a, (3) c. Let's test it. We already showed that pair (1) b is correct. For (2): \(f(x)=\sin(\tan^{-1}(x))\) and \(f^{-1}(x)=x/\sqrt{-x^2}\) It can also be seen here that the function f(x) and its inverse are not the same form. Thus, option (c) is not correct.

Testing Option (d)

Option (d) states (1) c, (2) b, (3) a. Since option (d) has only one new pair other than (1) b, it must be the correct answer. We check it anyway. For (3): \(f(x)=2x+3\) and \(f^{-1}(x)=\frac{(x-3)}{2}\) Substitute \(f^{-1}(x)\) in \(f(x)\): \[f(f^{-1}(x)) = 2\left(\frac{(x-3)}{2}\right)+3\] \[f(f^{-1}(x)) = x-3+3\] \[f(f^{-1}(x)) = x\] Since f(f^{-1}(x)) = x, option (d) is correct. The correct match is: (a) (1) c, (2) b, (3) a.

Key concepts

Function Composition

The concept of function composition involves combining two functions where the output of one function becomes the input of another. In symbolic terms, if you have functions \( f(x) \) and \( g(x) \), their composition \( f(g(x)) \) means you plug \( x \) into \( g(x) \) first and then take the result and input it into \( f(x) \). This concept is incredibly useful in mathematics for creating more complex functions from simpler ones, and it's likewise foundational in solving problems involving inverse functions.

Whenever you encounter a scenario where you're asked to verify inverse functions, you're essentially being asked to perform function composition. For a pair of functions \( f(x) \) and \( f^{-1}(x) \) to be inverses, the composition \( f(f^{-1}(x)) \) or \( f^{-1}(f(x)) \) must return the input \( x \). This verifies that every operation done by one function is undone by the other.

In the exercise, through substitution and calculation, we determine the match between functions and their proposed inverses. If the composition results back to the original input \( x \), the pair is correctly identified as function and its inverse.

Inverse Trigonometric Functions

Inverse trigonometric functions, like \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \), are the inverses of the original trigonometric functions, \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \) respectively. They are used to determine the angle that corresponds to a given trigonometric ratio.

Understanding inverse trigonometric functions requires knowing that these functions return angles. For example, if \( \sin(\theta) = x \), then \( \sin^{-1}(x) = \theta \), where \( \theta \) is an angle.

In the context of the exercise, the function \( f(x) = \sin(\tan^{-1}(x)) \) shows a combination of a trigonometric function and its inverse. Evaluating this involves understanding that \( \tan^{-1}(x) \) gives an angle whose tangent is \( x \), and then \( \sin() \) of that angle gives \( \sin(\tan^{-1}(x)) \). However, the inverse given in the problem, which is \( x / \sqrt{-x^2} \), does not match the required outcome for verification.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. In base-2, for example, the function \( \log_2(x) \) answers the question: "To what exponent must 2 be raised, to yield \( x \)?". This is a central concept in mathematics, allowing conversion between multiplicative and additive processes.

Logarithmic functions, like \( f(x) = \log_2(x) \), reverse what exponential functions do, simplifying the process of solving for unknowns in exponential equations.

• Key Component: They convert multiplication into addition, niched in simplifying equations.

In the given exercise, the function \( f(x) = 1 - 2^{-x} \) and the inverse \( f^{-1}(x) = -\log_2(1-x) \) are paired. Using the properties of logarithms and exponents, substituting \( f^{-1}(x) \) into \( f(x) \) yields \( x \) as expected, confirming the correctness of the inverse pair.

Mathematical Matching

Mathematical matching involves correctly pairing functions with their respective inverses or corresponding equations. This concept within inverse functions is critical for ensuring that the operations in a function are reversible and logically consistent.

To achieve correct matching, you perform a set of checks to verify each proposed inverse by using function composition. If substituting the inverse into the original function, and vice versa, results in the identity function (where you get back the input \( x \)), then the pair is correctly matched.

In this specific exercise, the process of testing different options through function composition is a demonstration of mathematical matching. By systematically substituting calculated inverses back into the functions and verifying that you consistently return to the original input, you ensure that each function is correctly matched with its inverse. The successful verification of option (d) demonstrates aptitude in mathematical matching, confirming that (1) \( c \), (2) \( b \), (3) \( a \) is the correct answer.