How do we find from its inverse function?

If , then . Replacing with after solving gives the original function .

Can we directly compare coefficients of and ?

Yes, because the equality must hold for all and are natural numbers.

JEE MainMathematics

Reviewed by official_key

+4 marks−1 if incorrectNumericalpyq

Sets, Relations and Functions: Mathematics | JEE Main

For some

a, b, c \in \mathbb{N}

, let

f(x) = ax - 3

and

g(x) = x^b + c, x \in \mathbb{R}

. If

(f \circ g)^{-1}(x) = \left(\frac{x-7}{2}\right)^{1/3}

, then

(f \circ g)(ac) + (g \circ f)(b)

is equal to

Your answer stays private

What feels right?

No score. Commit to your first instinct. We’ll show what your mind noticed and what it missed.

Type the value - units or words beside it are fine.

Step-by-step solutionView

Correct answer

The value of $(f \circ g)(ac) + (g \circ f)(b)$ is 2039.

Option analysis

Why each option works or fails

Reviewed route

Solution

StepWorking

01given

$a, b, c \in \mathbb{N}$ , $f(x) = ax - 3$ , $g(x) = x^b + c$ , and $(f \circ g)^{-1}(x) = \left(\frac{x-7}{2}\right)^{1/3}$ .

02goal

Find the value of $(f \circ g)(ac) + (g \circ f)(b)$ .

03approach

Invert $(f \circ g)^{-1}(x)$ to find the explicit formula for $(f \circ g)(x)$ . Then equate it to $f(g(x)) = a(x^b + c) - 3$ to determine the natural numbers $a, b, c$ . Finally, evaluate $(f \circ g)(ac)$ and $(g \circ f)(b)$ and compute their sum.

04execute

Let $y = (f \circ g)^{-1}(x) = \left(\frac{x-7}{2}\right)^{1/3}$ . Inverting gives $y^3 = \frac{x-7}{2} \implies x = 2y^3 + 7$ . Hence, $(f \circ g)(x) = 2x^3 + 7$ .

05execute

Compute $f(g(x))$ from definitions: $f(g(x)) = a(x^b + c) - 3 = a x^b + ac - 3$ . Equating this to $2x^3 + 7$ for all $x$ : $a x^b + (ac - 3) = 2x^3 + 7$ . By comparing coefficients and exponents: $a = 2$ , $b = 3$ , and $ac - 3 = 7 \implies 2c = 10 \implies c = 5$ . Since $a, b, c \in \mathbb{N}$ , these values are valid. Then $ac = 2 \times 5 = 10$ .

06execute

Calculate the two terms: First, $(f \circ g)(ac) = (f \circ g)(10) = 2(10)^3 + 7 = 2(1000) + 7 = 2007$ . Second, evaluate $(g \circ f)(b) = g(f(3))$ . Since $f(3) = a(3) - 3 = 2(3) - 3 = 3$ , we have $g(f(3)) = g(3) = 3^b + c = 3^3 + 5 = 27 + 5 = 32$ .

07execute

Sum the two values: $(f \circ g)(ac) + (g \circ f)(b) = 2007 + 32 = 2039$ .

✓verify

Check: $f(x) = 2x - 3$ , $g(x) = x^3 + 5$ . $(f \circ g)(x) = 2(x^3 + 5) - 3 = 2x^3 + 7$ . Its inverse is $y = 2x^3 + 7 \implies x = \left(\frac{y-7}{2}\right)^{1/3}$ , which matches the stem exactly.

Hints that build this answer step by step

What is the explicit expression for the composition $(f \circ g)(x)$ in terms of $a, b,$ and $c$ ?
$(f \circ g)(x) = a(x^b + c) - 3 = a x^b + ac - 3$
Given $(f \circ g)^{-1}(x) = \left(\frac{x - 7}{2}\right)^{1/3}$ , what are the values of the natural numbers $a, b,$ and $c$ ?
$a = 2, b = 3, c = 5$
Using $a = 2, b = 3, c = 5$ , what is the numerical value of $(f \circ g)(ac) + (g \circ f)(b)$ ?
2039

✓ Source and academic review

Question type: Numerical
Exam relevance: JEE Main · Mathematics
Concepts assessed: Mathematics
Academic status: Reviewed by official_key
Source: pyq
Editorial review: 9 September 2026

Quick checks

Students also ask

How do we find

f(g(x))

from its inverse function?

If $h^{-1}(x) = y$ , then $h(y) = x$ . Replacing $y$ with $x$ after solving gives the original function $h(x) = (f \circ g)(x)$ .

Can we directly compare coefficients of

a x^b

and

2 x^3

Yes, because the equality $a x^b + ac - 3 = 2x^3 + 7$ must hold for all $x \in \mathbb{R}$ and $a, b, c$ are natural numbers.

Answer

The value of $(f \circ g)(ac) + (g \circ f)(b)$ is 2039.

Why each option works or fails

Step-by-step solution

given: $a, b, c \in \mathbb{N}$ , $f(x) = ax - 3$ , $g(x) = x^b + c$ , and $(f \circ g)^{-1}(x) = \left(\frac{x-7}{2}\right)^{1/3}$ .
goal: Find the value of $(f \circ g)(ac) + (g \circ f)(b)$ .
approach: Invert $(f \circ g)^{-1}(x)$ to find the explicit formula for $(f \circ g)(x)$ . Then equate it to $f(g(x)) = a(x^b + c) - 3$ to determine the natural numbers $a, b, c$ . Finally, evaluate $(f \circ g)(ac)$ and $(g \circ f)(b)$ and compute their sum.
execute: Let $y = (f \circ g)^{-1}(x) = \left(\frac{x-7}{2}\right)^{1/3}$ . Inverting gives $y^3 = \frac{x-7}{2} \implies x = 2y^3 + 7$ . Hence, $(f \circ g)(x) = 2x^3 + 7$ .
execute: Compute $f(g(x))$ from definitions: $f(g(x)) = a(x^b + c) - 3 = a x^b + ac - 3$ . Equating this to $2x^3 + 7$ for all $x$ : $a x^b + (ac - 3) = 2x^3 + 7$ . By comparing coefficients and exponents: $a = 2$ , $b = 3$ , and $ac - 3 = 7 \implies 2c = 10 \implies c = 5$ . Since $a, b, c \in \mathbb{N}$ , these values are valid. Then $ac = 2 \times 5 = 10$ .
execute: Calculate the two terms: First, $(f \circ g)(ac) = (f \circ g)(10) = 2(10)^3 + 7 = 2(1000) + 7 = 2007$ . Second, evaluate $(g \circ f)(b) = g(f(3))$ . Since $f(3) = a(3) - 3 = 2(3) - 3 = 3$ , we have $g(f(3)) = g(3) = 3^b + c = 3^3 + 5 = 27 + 5 = 32$ .
execute: Sum the two values: $(f \circ g)(ac) + (g \circ f)(b) = 2007 + 32 = 2039$ .
verify: Check: $f(x) = 2x - 3$ , $g(x) = x^3 + 5$ . $(f \circ g)(x) = 2(x^3 + 5) - 3 = 2x^3 + 7$ . Its inverse is $y = 2x^3 + 7 \implies x = \left(\frac{y-7}{2}\right)^{1/3}$ , which matches the stem exactly.

Sets, Relations and Functions: Mathematics | JEE Main

What feels right?

The value of $(f \circ g)(ac) + (g \circ f)(b)$ is 2039.

Why each option works or fails

Solution

01given

02goal

03approach

04execute

05execute

06execute

07execute

✓verify

We think you should solve this next

Let R be a relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a, b)\text{R}(c, d)$ i…

If the domain of the function $f(x) = \frac{[x]}{1+x^2}$ , where $[x]$ is greatest integer…

Suppose $f: \mathbb{R} \rightarrow (0, \infty)$ be a differentiable function such that…

Students also ask

Sets, Relations and Functions: Mathematics | JEE Main

The value of (f∘g)(ac)+(g∘f)(b)(f \circ g)(ac) + (g \circ f)(b)(f∘g)(ac)+(g∘f)(b) is 2039.

01given

02goal

03approach

04execute

05execute

06execute

07execute

✓verify

Let R be a relation on N×N\mathbb{N} \times \mathbb{N}N×N defined by (a,b)R(c,d)(a, b)\text{R}(c, d)(a,b)R(c,d) i…

If the domain of the function f(x)=[x]1+x2f(x) = \frac{[x]}{1+x^2}f(x)=1+x2[x]​, where [x][x][x] is greatest integer…

Suppose f:R→(0,∞)f: \mathbb{R} \rightarrow (0, \infty)f:R→(0,∞) be a differentiable function such that…

Students also ask

The value of $(f \circ g)(ac) + (g \circ f)(b)$ is 2039.

Let R be a relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a, b)\text{R}(c, d)$ i…

If the domain of the function $f(x) = \frac{[x]}{1+x^2}$ , where $[x]$ is greatest integer…

Suppose $f: \mathbb{R} \rightarrow (0, \infty)$ be a differentiable function such that…