Worksheets and Answers: SU23 wuw-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNslkr7j8P3jH-IBIWJZaoC-hHsOMixm1cv7ZvsXq5H9X-1NBZHajmi GRAPHS AND FUNCTIONS Combining functions: Advanced Suppose that the functions $f$ and $g$ are defined as follows. \[ \begin{array}{l} f(x)=5 x^{2}-3 \\ g(x)=\sqrt{3 x-1} \end{array} \] Find $f+g$ and $f \cdot g$. Then, give their domains using interval notation. \[ (f+g)(x)= \] Domain of $f+g$ : \[ (f \cdot g)(x)= \] Domain of $f \cdot g$ : Explanation Check $75^{\circ} \mathrm{F}$ Clear Search (b)

Step 1 ：The question is asking for two things. First, it wants the sum of the functions $f$ and $g$, denoted as $(f+g)(x)$. Second, it wants the product of the functions $f$ and $g$, denoted as $(f \cdot g)(x)$. Additionally, it wants the domain of both these new functions.

Step 2 ：The domain of a function is the set of all possible input values (x-values) which will output real numbers.

Step 3 ：To find $(f+g)(x)$, we simply add the two functions together. To find $(f \cdot g)(x)$, we multiply the two functions together.

Step 4 ：The domain of $(f+g)(x)$ and $(f \cdot g)(x)$ will be the intersection of the domains of $f$ and $g$. The domain of $f(x)$ is all real numbers since any real number can be squared, multiplied by 5, and subtracted by 3. The domain of $g(x)$ is $x \geq \frac{1}{3}$ because the inside of the square root must be greater than or equal to 0 (we can't take the square root of a negative number).

Step 5 ：So, the domain of both $(f+g)(x)$ and $(f \cdot g)(x)$ will be $x \geq \frac{1}{3}$.

Step 6 ：\[ (f+g)(x) = 5x^2 + \sqrt{3x - 1} - 3 \]

Step 7 ：Domain of $(f+g)$ : $x \geq \frac{1}{3}$

Step 8 ：\[ (f \cdot g)(x) = (5x^2 - 3) \cdot \sqrt{3x - 1} \]

Step 9 ：Domain of $(f \cdot g)$ : $x \geq \frac{1}{3}$

Step 10 ：\boxed{\text{Final Answer: }}

Step 11 ：\boxed{\text{The function }} (f+g)(x) \text{ is } 5x^2 + \sqrt{3x - 1} - 3 \text{ and its domain is } x \geq \frac{1}{3}.

Step 12 ：\boxed{\text{The function }} (f \cdot g)(x) \text{ is } (5x^2 - 3) \cdot \sqrt{3x - 1} \text{ and its domain is also } x \geq \frac{1}{3}.

Step 13 ：\boxed{\text{So, the final answers are: }}

Step 14 ：\boxed{(f+g)(x) = 5x^2 + \sqrt{3x - 1} - 3}

Step 15 ：\boxed{\text{Domain of }} (f+g) : x \geq \frac{1}{3}

Step 16 ：\boxed{(f \cdot g)(x) = (5x^2 - 3) \cdot \sqrt{3x - 1}}

Step 17 ：\boxed{\text{Domain of }} (f \cdot g) : x \geq \frac{1}{3}