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 , we need to compute , where and . Substituting into , we have:
Step 1: Expand
= 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.Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Function Composition: Solve (f ∘ g)(x) and (g ∘ f)(x) with f(x) = -3x^2 + 1 and g(x) = 3x
How to Find g(f(x)) for f(x) = 2x^2 + 3 and g(x) = 1/x
Determining g(x) in Function Composition f(x) = x - 3 and g(f(x)) = 5 / (x² - 6x + 5)
Function Composition: Find (f ∘ g)(x) and (g ∘ f)(x) with f(x) = 3x² + 5 and g(x) = -3x
Function Composition: Solving f(g(5)) and g(f(x))