Problem 17 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f: R \rightarrow R, g: R \rightarrow R\) be two functions given by \(f(x)=2 x-3, g(x)=x^{3}+5\). Then \((f o g)^{-1}(x)\) is equal to (A) \(\left(\frac{x-7}{2}\right)^{1}\) (B) \(\left(\frac{x+7}{2}\right)^{1}\) (C) \(\left(x-\frac{7}{2}\right)^{1 / 3}\) (D) \(\left(\frac{x-2}{7}\right)^{1 / 3}\)

Step-by-Step Solution

Answer

Option (A) \(\left(\frac{x-7}{2}\right)^{1/3}\) is correct.

Step 1: Understanding Composition

To find \((f \circ g)(x)\), we need to substitute \(g(x)\) into \(f(x)\). So, we compute \((f \circ g)(x) = f(g(x))\).

Step 2: Substitute and Simplify

Substitute \(g(x) = x^3 + 5\) into \(f(x) = 2x - 3\). This gives us \[ (f \circ g)(x) = 2(x^3 + 5) - 3 \]. Simplifying this, we have \(2x^3 + 10 - 3\), which further simplifies to \(2x^3 + 7\). So, \((f \circ g)(x) = 2x^3 + 7\).

Step 3: Find the Inverse

Now, we need to find the inverse of the function \((f \circ g)(x) = 2x^3 + 7\). To find the inverse, we set \(y = 2x^3 + 7\) and solve for \(x\).

Step 4: Solve for x

Starting with the equation \(y = 2x^3 + 7\), we first subtract 7 from both sides to get \(y - 7 = 2x^3\). Then, divide both sides by 2 to isolate \(x^3\), resulting in \(\frac{y - 7}{2} = x^3\). Finally, take the cube root of both sides to find \(x\), so \(x = \left(\frac{y - 7}{2}\right)^{1/3}\).

Step 5: Conclusion

The expression \(\left(\frac{y-7}{2}\right)^{1/3}\) is the inverse function, so \((f \circ g)^{-1}(x) = \left(\frac{x-7}{2}\right)^{1/3}\). Compare this to the given options.

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics that involves combining two functions into a single function. If you have two functions, say \(f(x)\) and \(g(x)\), the composition of these functions, written as \((f \circ g)(x)\), is essentially finding \(f(g(x))\). In this process, you take the output of \(g(x)\) and use it as the input for \(f(x)\).

Here, for example, given \(f(x) = 2x - 3\) and \(g(x) = x^3 + 5\), the composition \((f \circ g)(x)\) is calculated by substituting \(g(x)\) into \(f(x)\). This involves replacing every occurrence of \(x\) in \(f(x)\) with \(x^3 + 5\).

This operation demonstrates how two functions can interact seamlessly to produce a new function, which can then be analyzed for various characteristics, such as finding its inverse.

Algebraic Manipulation

Algebraic manipulation is about using algebraic techniques to simplify or rearrange expressions. Once we have the composed function \((f \circ g)(x) = 2x^3 + 7\), we use algebraic manipulation to express it in a more digestible form, if possible.

Starting from \(2(x^3 + 5) - 3\), simple arithmetic and distributive properties help us simplify: multiplying through the term inside the parentheses by 2 and then subtracting 3. This results in the expression \(2x^3 + 10 - 3\) simplifying to \(2x^3 + 7\).

This step-by-step breakdown of 'substitute and simplify' is a key aspect of algebraic manipulation. These processes allow us to work more straightforwardly with complex functions, preparing them for further operations like finding inverses.

Finding Inverses

Finding inverses of functions reverses the process of function application. When looking for the inverse of a function such as \((f \circ g)(x) = 2x^3 + 7\), we aim to find another function that will return the input \(x\) when \((f \circ g)\)'s output is plugged into it.

To find the inverse, begin by setting \(y = 2x^3 + 7\). The goal is to solve this equation for \(x\):

• Subtract 7 from both sides to isolate terms involving \(x\): \(y - 7 = 2x^3\).
• Divide each side by 2: \(\frac{y - 7}{2} = x^3\).
• Finally, apply the cube root to both sides to solve for \(x\), yielding \(x = \left(\frac{y - 7}{2}\right)^{1/3}\).

Therefore, the function \(x = \left(\frac{x - 7}{2}\right)^{1/3}\) represents the inverse, denoted as \((f \circ g)^{-1}(x)\). This final expression allows us to retrace our steps from the function's output back to its original input.