Question: Given \( f(x) = 5x + 3 \) and \( g(x) = 3x^2 \), first find \( f + g \), \( f - g \), \(…

Given \( f(x) = 5x + 3 \) and \( g(x) = 3x^2 \), first find \( f + g \), \( f - g \), \( fg \), and \( \frac{f}{g} \). Then determine the domain for each function.

\[ (f + g)(x) = 5x + 3 + 3x^2 \quad \text{(Simplify your answer.)} \]

What is the domain of \( f + g \)?

A. \([0, \infty)\)

B. \(\left(-\infty, -\frac{1}{2}\right) \cup \left(-\frac{1}{2}, \infty\right)\)

C. \((-\infty, \infty)\)

D. \(\left(-\frac{1}{2}, \infty\right)\)

\[ (f - g)(x) = \boxed{\phantom{xxxx}} \quad \text{(Simplify your answer.)} \]

Solution

Given \( f(x) = 5x + 3 \) and \( g(x) = 3x^2 \), we need to find \( f+g \), \( f-g \), \( fg \), and \( \frac{f}{g} \), and then determine the domain for each function. 1. Finding \( f+g \): \[ f(x) + g(x) = (5x + 3) + 3x^2 \] Simplify the expression: \[ = 3x^2 + 5x + 3 \] 2. Finding the domain of \( f+g \): Both \( f(x) \) and \( g(x) \) are polynomials. The domain of a polynomial is all real numbers. So, the domain of \( f+g \) is: \[ (-\infty, \infty) \] 3. Finding \( f-g \): \[ f(x) - g(x) = (5x + 3) - 3x^2 \] Simplify the expression: \[ = -3x^2 + 5x + 3 \] 4. Finding the domain of \( f-g \): The domain is also all real numbers since both \( f(x) \) and \( g(x) \) are polynomials. So, the domain of \( f-g \) is: \[ (-\infty, \infty) \] 5. Finding \( fg \): \[ f(x) \cdot g(x) = (5x + 3)(3x^2) \] Simplify the expression: \[ = 15x^3 + 9x^2 \] 6. Finding the domain of \( fg \): The domain is all real numbers: \[ (-\infty, \infty) \] 7. Finding \( \frac{f}{g} \): \[ \frac{f(x)}{g(x)} = \frac{5x + 3}{3x^2} \] Identify division by zero: The denominator \( 3x^2 = 0 \) when \( x = 0 \). So, the domain of \( \frac{f}{g} \) is all real numbers except \( x = 0 \): \[ (-\infty, 0) \cup (0, \infty) \]