Question

Let f : R→R be defined by f(x)= 3x²-5 and g : R→R by g(x)=\(\frac{x}{x^2+1}\) then gof is

• A. \(\frac{3x^2}{x^4+2x^2-4}\)
• B. \(\frac{3x^2-5}{9x^4-30x^2-26}\)
• C. \(\frac{3x^2}{9x^4+30x^2-2}\)
• D. \(\frac{3x^2-5}{9x^4-6x^2+26}\)

Hint:

When dealing with composite functions, remember to substitute the output of the first function into the second. For example, for \( (g \circ f)(x) \), substitute \( f(x) \) into \( g(x) \), and then simplify the resulting expression. Pay attention to the algebraic simplifications in the denominator when squaring terms.

Answer 1

The Correct Option is B

The correct answer is (B) : \(\frac{3x^2-5}{9x^4-30x^2-26}\).

Answer 2

The correct answer is: (B) \( \frac{3x^2 - 5}{9x^4 - 30x^2 - 26} \) .

We are given two functions:

• f(x) = 3x² - 5 (a function from \( \mathbb{R} \) to \( \mathbb{R} \))
• g(x) = \( \frac{x}{x^2 + 1} \) (a function from \( \mathbb{R} \) to \( \mathbb{R} \))

We need to find the composite function \( (g \circ f)(x) \), which means applying \( f(x) \) first, and then applying \( g(x) \) to the result of \( f(x) \). The composite function \( (g \circ f)(x) \) is given by:

\( g(f(x)) = g(3x^2 - 5) \)

Now substitute \( f(x) \) into \( g(x) \):

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

Now simplify the denominator:

\( (3x^2 - 5)^2 + 1 = 9x^4 - 30x^2 + 25 + 1 = 9x^4 - 30x^2 - 26 \)

Therefore, the composite function is:

\( g(f(x)) = \frac{3x^2 - 5}{9x^4 - 30x^2 - 26} \)

Thus, the correct answer is (B) \( \frac{3x^2 - 5}{9x^4 - 30x^2 - 26} \).