Lambert Fitzgerald
06/12/2024 · Junior High School
The functions \( f \) and \( a \) are defined as \( f(x)=4 x-3 \) and \( a(x)=-9 x^{2} \). The domain of \( f f \) is \( (-\infty, \infty) \). (Type your answer in interval notation.) The domain of \( \left(\frac{f}{g}\right)(x) \) is \( (-\infty, 0) \cup(0, \infty) \). (Type your answer in interval notation.) The domain of \( \left(\frac{g}{f}\right)(x) \) is \( \left(-\infty, \frac{3}{4}\right) \cup\left(\frac{3}{4}, \infty\right) \). (Type your answer in interval notation.) b) \( (f+g)(x)=-9 x^{2}+4 x-3 \) (Simplify your answer.) ( \( f-g)(x)=\square \) (Simplify your answer.)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To solve the problem, we first need to clarify the functions involved. We have:
- \( f(x) = 4x - 3 \)
- \( a(x) = -9x^2 \)
However, it seems there is a function \( g(x) \) that is not explicitly defined in your question. We will assume \( g(x) = a(x) = -9x^2 \) for the purpose of this problem.
### Part b: Simplifying \( (f - g)(x) \)
1. **Finding \( (f + g)(x) \)**:
\[
(f + g)(x) = f(x) + g(x) = (4x - 3) + (-9x^2)
\]
\[
= -9x^2 + 4x - 3
\]
2. **Finding \( (f - g)(x) \)**:
\[
(f - g)(x) = f(x) - g(x) = (4x - 3) - (-9x^2)
\]
\[
= 4x - 3 + 9x^2
\]
\[
= 9x^2 + 4x - 3
\]
Thus, the simplified expression for \( (f - g)(x) \) is:
\[
(f - g)(x) = 9x^2 + 4x - 3
\]
Quick Answer
\( (f - g)(x) = 9x^2 + 4x - 3 \)
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