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QuestionFind the domain and range of f(x)=3+2xf(x)=\sqrt{3+2x} and the domain and range of its inverse.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=3+xf(x)=\sqrt{3+x} . We are looking for the domain and range of the function and its inverse

STEP 2

First, let's find the domain of the function. The domain of a function is the set of all possible values of xx that will give real numbers when substituted into the function. For a square root function, the expression inside the square root must be greater than or equal to zero.
So, we have+2x≥0 +2x \geq0

STEP 3

Next, solve the inequality for xx.
2x≥−32x \geq -3x≥−32x \geq -\frac{3}{2}So, the domain of the function f(x)f(x) is [−3/2,∞)[-3/2, \infty).

STEP 4

Now, let's find the range of the function. The range of a function is the set of all possible values of f(x)f(x) that we get after substituting all the possible values of xx from the domain into the function. Since the square root function always gives non-negative values, the range of the function f(x)f(x) is [0,∞)[0, \infty).

STEP 5

Next, let's find the domain of the inverse function. The domain of the inverse function is the same as the range of the original function. So, the domain of the inverse function is [0,∞)[0, \infty).

STEP 6

Lastly, let's find the range of the inverse function. The range of the inverse function is the same as the domain of the original function. So, the range of the inverse function is [−3/2,∞)[-3/2, \infty).
So, the domain of the function is [−3/2,∞)[-3/2, \infty), the range of the function is [0,∞)[0, \infty), the domain of the inverse is [0,∞)[0, \infty), and the range of the inverse is [−3/2,∞)[-3/2, \infty).

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