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f(x) =sqrt 3x-5

What are the domain and range of F
Find f^-1 (x)
Please show all steps to prove thank you

Domain of f(x): x ≥ 5/√3

Range of f(x): All real numbers

f^-1(x) = (x + 5)/√3

Did you know?

Did you know that the domain and range of a function can affect its behavior and the values it can take? In the given function f(x) = √(3x - 5), the domain represents all the possible values of x for which the function is defined and the range represents all the possible values of f(x). To find the domain, we need to determine the values of x that make the expression inside the square root non-negative. Therefore, we set 3x - 5 ≥ 0, and by solving this inequality, we find x ≥ 5/3. Hence, the domain of f(x) is all real numbers greater than or equal to 5/3. To find the range, we can consider the behavior of the square root function. Since the square root of any non-negative number is always positive or zero, the expression 3x - 5 must be non-negative as well. This implies that 3x - 5 ≥ 0, which leads to x ≥ 5/3. As the function is increasing for x ≥ 5/3, the smallest value f(x) can take is f(5/3) = 0. Hence, the range of f(x) is all real numbers greater than or equal to 0. Now, to find f^-1(x), we are looking for the inverse function of f(x). To solve this, we need to swap the roles of x and f(x) and solve for x. Step 1: Replace f(x) with y. y = √(3x - 5) Step 2: Swap x and y. x = √(3y - 5) Step 3: Isolate y. x^2 = 3y - 5 3y = x^2 + 5 y = (x^2 + 5)/3 The inverse function f^-1(x) is given by (x^2 + 5)/3. By following these steps, we have successfully determined the domain and range of f(x) and found the inverse function f^-1(x). Understanding these concepts helps to analyze and comprehend the behavior of functions.