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QuestionFind the domain and range of f(x)=3+2xf(x)=\sqrt{3+2x} and its inverse f−1(x)f^{-1}(x).

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=3+xf(x)=\sqrt{3+x}

STEP 2

First, we need to determine the domain of the function f(x)f(x). The domain is the set of all possible x-values that can be plugged into the function. For the square root function, the value inside the square root (the radicand) must be greater than or equal to zero, because we cannot take the square root of a negative number in the real number system.
+2x≥0+2x \geq0

STEP 3

Now, solve the inequality to find the domain of the function.
2x≥−32x \geq -3

STEP 4

Divide each side of the inequality by2.
x≥−32x \geq -\frac{3}{2}

STEP 5

So, the domain of the function f(x)f(x) is {x∣x≥−32}\{x \mid x \geq -\frac{3}{2}\}.

STEP 6

Next, we need to determine the range of the function f(x)f(x). The range is the set of all possible y-values that the function can output. Since the function is a square root function, the output (y-values) will always be greater than or equal to zero.
So, the range of the function f(x)f(x) is {y∣y≥0}\{y \mid y \geq0\}.

STEP 7

Now, let's find the domain and range of the inverse of the function f(x)f(x). First, we need to find the inverse function f−1(x)f^{-1}(x). To do this, we swap x and y in the original function and solve for y.
So, the inverse function is f−1(x)=x2−32f^{-1}(x)=\frac{x^2-3}{2}.

STEP 8

The domain of the inverse function f−1(x)f^{-1}(x) is the range of the original function f(x)f(x). So, the domain of f−1(x)f^{-1}(x) is {x∣x≥0}\{x \mid x \geq0\}.

STEP 9

The range of the inverse function f−(x)f^{-}(x) is the domain of the original function f(x)f(x). So, the range of f−(x)f^{-}(x) is {y∣y≥−32}\{y \mid y \geq -\frac{3}{2}\}.
In conclusion, for the function f(x)=3+2xf(x)=\sqrt{3+2x}i) the domain is {x∣x≥−32}\{x \mid x \geq -\frac{3}{2}\}, ii) the range is {y∣y≥}\{y \mid y \geq\}, iii) the domain of the inverse is {x∣x≥}\{x \mid x \geq\}, iv) the range of the inverse is {y∣y≥−32}\{y \mid y \geq -\frac{3}{2}\}.

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