Q. If \( f(x) = 3x + 2 \) and \( g(x) = x^2 – x \), Find the Value

Answer

Identify the functions.
Let f(x) = 3x + 2 and g(x) = x^2 – x.
Function composition.
If finding f(g(x)), or f composed with g:

\[ f(g(x)) = 3(x^2 – x) + 2 = 3x^2 – 3x + 2 \]
Alternate composition.
If finding g(f(x)), or g composed with f:

\[ g(f(x)) = (3x + 2)^2 – (3x + 2) \]

\[ 9x^2 + 12x + 4 – 3x – 2 = 9x^2 + 9x + 2 \]

Detailed Explanation

Solution

Understand the given functions.
f(x) = 3x + 2 and g(x) = x^2 – x.
Find the composition f(g(x)).
Substitute g(x) into f(x).

\[ f(g(x)) = 3(x^2 – x) + 2 \]

\[ 3x^2 – 3x + 2 \]
Find the composition g(f(x)).
Substitute f(x) into g(x).

\[ g(f(x)) = (3x + 2)^2 – (3x + 2) \]

Expand the squared term:

\[ 9x^2 + 12x + 4 – 3x – 2 \]

\[ 9x^2 + 9x + 2 \]

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Frequently Asked Questions

What is (f + g)(x)?

(f + g)(x) = (3x + 2) + (x^2 - x) = x^2 + 2x + 2.

What is (f * g)(x)?

(f * g)(x) = (3x + 2)(x^2 - x) = 3x^3 - x^2 - 2x.

What is (f / g)(x) and its domain?

(f / g)(x) = (3x + 2)/(x^2 - x). Domain: all real x except where x^2 - x = 0, so x ≠ 0 and x ≠ 1.

What is the composition f(g(x))?

f(g(x)) = 3(x^2 - x) + 2 = 3x^2 - 3x + 2.

What is the composition g(f(x))?

g(f(x)) = (3x + 2)^2 - (3x + 2) = 9x^2 + 9x + 2.

How do I find the inverse of f?

For f(x) = 3x + 2, swap x and y then solve: y = (x - 2)/3. So f^{-1}(x) = (x - 2)/3.

What are the domain and range of f and g?

f: domain all real numbers, range all real numbers. g: domain all real numbers; vertex at x = 1/2 gives minimum g(1/2) = -1/4, so range is [-1/4, ∞).

What is f(g(2)) numerically?

g(2) = 2, so f(g(2)) = f(2) = 3(2)+2 = 8.

Could f and g be added, subtracted, or composed in any order?

Yes: you can add, subtract, multiply, divide (except where denominator zero), and compose either f∘g or g∘f; results differ because composition is not commutative.

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