For x R, two real valued functions f(x) and g(x) are such that, g(x) = x + 1 and f g(x) = x + 3 - x. Then f(0) is equal to:

Given: g(x) = x + 1 and f g(x) = x + 3 - x. Find: f(0). Let u = g(x) = x + 1 Then x = u - 1 and x = (u-1)^2 Now f(g(x)) = x + 3 - x becomes f(u) = (u-1)^2 + 3 - (u-1) Simplifying, aligned f(u) &= u^2 - 2u + 1 + 3 - u + 1 &= u^2 - 3u + 5 aligned Hence, f(x) = x^2 - 3x + 5 Therefore, f(0) = 0^2 - 3 0 + 5 = 5 So, the correct option is A. Given: g(x) = x + 1. Find: a function f consistent with the composition. Take f(u) = u^2 - 3u + 5 Then substituting u = g(x) = x + 1, f(g(x)) = ( x+1)^2 - 3( x+1) + 5 Now expand: aligned f(g(x)) &= x + 2 x + 1 - 3 x - 3 + 5 &= x - x + 3 aligned This exactly matches the given expression x + 3 - x. Hence, f(x) = x^2 - 3x + 5 and therefore f(0) = 5

For x R, two real valued functions f(x) and g(x) are such that, g… — Solution | JEE Mathematics

MCQMediumJEE 2023Composition & Inverse Functions

For $x \in \mathbb{R}$ , two real valued functions $f(x)$ and $g(x)$ are such that,

g(x) = \sqrt{x} + 1 \quad \text{and} \quad f \circ g(x) = x + 3 - \sqrt{x}.

Then $f(0)$ is equal to:

A
$5$
B
$0$
C
$-3$
D
$1$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: $g(x) = \sqrt{x} + 1$ and $f \circ g(x) = x + 3 - \sqrt{x}$ .

Find: $f(0)$ .

Let

u = g(x) = \sqrt{x} + 1

Then

\sqrt{x} = u - 1 \quad \text{and} \quad x = (u-1)^2

Now

f(g(x)) = x + 3 - \sqrt{x}

becomes

f(u) = (u-1)^2 + 3 - (u-1)

Simplifying,

\begin{aligned} f(u) &= u^2 - 2u + 1 + 3 - u + 1 \\ &= u^2 - 3u + 5 \end{aligned}

Hence,

f(x) = x^2 - 3x + 5

Therefore,

f(0) = 0^2 - 3 \cdot 0 + 5 = 5

So, the correct option is A.

Verification from the given composition

Given: $g(x) = \sqrt{x} + 1$ .

Find: a function $f$ consistent with the composition.

Take

f(u) = u^2 - 3u + 5

Then substituting $u = g(x) = \sqrt{x} + 1$ ,

f(g(x)) = (\sqrt{x}+1)^2 - 3(\sqrt{x}+1) + 5

Now expand:

\begin{aligned} f(g(x)) &= x + 2\sqrt{x} + 1 - 3\sqrt{x} - 3 + 5 \\ &= x - \sqrt{x} + 3 \end{aligned}

This exactly matches the given expression $x + 3 - \sqrt{x}$ . Hence,

f(x) = x^2 - 3x + 5

and therefore

f(0) = 5

Common mistakes

Assuming directly that $f(x) = x + 3 - \sqrt{x}$ is incorrect because that expression is for $f(g(x))$ , not for $f(x)$ itself. First rewrite the input of $f$ as a new variable.
Using $u = \sqrt{x}$ instead of $u = \sqrt{x} + 1$ leads to the wrong substitution. The full expression $g(x)$ must be treated as the argument of $f$ .
Forgetting to express $x$ in terms of $u$ causes an incomplete derivation. After taking $u = \sqrt{x} + 1$ , use $\sqrt{x} = u - 1$ and then $x = (u-1)^2$ .

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Answer

Step-by-step solution

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