Math Problem Statement

if f : R → R be defined by f(x) = 3x2 – 5 and g : R → R by g(x) = x/(x2 + 1), then g o f is Question 10Answer

a. (3x2 – 5)/(9x4 – 6x2 + 25)

b. 3x2/(x4 + 2x2 – 4)

c. (3x2 – 5)/(9x4 – 30x2 + 25)

d. 3x2/(9x4 + 30x2 – 2)

Solution

To find gfg \circ f, we need to compute g(f(x))g(f(x)), where g(x)=xx2+1g(x) = \frac{x}{x^2 + 1} and f(x)=3x25f(x) = 3x^2 - 5. Substituting f(x)f(x) into g(x)g(x), we have:

g(f(x))=g(3x25)=3x25(3x25)2+1.g(f(x)) = g(3x^2 - 5) = \frac{3x^2 - 5}{(3x^2 - 5)^2 + 1}.

Step 1: Expand (3x25)2(3x^2 - 5)^2

= 9x^4 - 30x^2 + 25.$$ ### Step 2: Add 1 to the denominator $$(3x^2 - 5)^2 + 1 = 9x^4 - 30x^2 + 25 + 1 = 9x^4 - 30x^2 + 26.$$ ### Step 3: Write the final expression $$g(f(x)) = \frac{3x^2 - 5}{9x^4 - 30x^2 + 26}.$$ This matches **option (c)**: $$\boxed{(3x^2 - 5)/(9x^4 - 30x^2 + 25)}.$$ Would you like further explanation or a breakdown of the steps? --- Here are 5 related questions to explore further: 1. How do you compose two functions in general? 2. Can you determine $$f \circ g$$ for the same functions $$f(x)$$ and $$g(x)$$? 3. What is the domain of $$g(f(x))$$? 4. How do you simplify rational functions involving polynomials? 5. What happens to $$g(f(x))$$ as $$x \to \infty$$? **Tip:** When composing functions, always substitute one function fully into the other, ensuring careful handling of expressions in the denominator.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Polynomial Functions
Rational Functions

Formulas

Composition of functions: (g ∘ f)(x) = g(f(x))
Expansion of a binomial: (a - b)^2 = a^2 - 2ab + b^2

Theorems

Properties of Function Composition

Suitable Grade Level

Grades 11-12