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Worksheets and Answers: SU23
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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 $f \cdot g$ :
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Answer

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

Steps

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}

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