QuestionFind the domain and $(f+g)(x)$ for $f(x)=\sqrt{2 x}$ and $g(x)=3 x-2$ . Simplify your answer.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x)=\sqrt{x}$ . The function $g(x)$ is defined as $g(x)=3x-$
3. The domain of a function is the set of all possible input values (x-values) which will produce a valid output from a particular function.

STEP 2

The sum of two functions, $(f+g)(x)$ , is defined as the sum of the outputs of $f(x)$ and $g(x)$ .
$(f+g)(x) = f(x) + g(x)$

STEP 3

Now, plug in the given functions $f(x)$ and $g(x)$ into the equation.
$(f+g)(x) = \sqrt{2x} + (3x -2)$

STEP 4

This is the simplified form of $(f+g)(x)$ . The domain of this function is the set of all x-values that make the expression defined.The square root function $\sqrt{2x}$ is defined for $x \geq0$ (since we can't take the square root of a negative number in the real number system), and the linear function $3x -2$ is defined for all real numbers. Therefore, the domain of $(f+g)(x)$ is $x \geq0$ .
So, the solution to part (a) is $(f+g)(x) = \sqrt{2x} +3x -2, \quad x \geq0$

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