Find the inverse of the function. f(x)=12?5×3
The correct answer and explanation is:
To find the inverse of the function f(x)=125x3f(x) = \frac{12}{5x^3}, follow these steps:
Step 1: Replace f(x)f(x) with yy
y=125x3y = \frac{12}{5x^3}
Step 2: Swap xx and yy
x=125y3x = \frac{12}{5y^3}
Step 3: Solve for yy
First, multiply both sides by 5y35y^3 to eliminate the denominator: 5xy3=125xy^3 = 12
Now divide both sides by 5x5x: y3=125xy^3 = \frac{12}{5x}
Take the cube root of both sides: y=125x3y = \sqrt[3]{\frac{12}{5x}}
Final Answer:
f−1(x)=125x3f^{-1}(x) = \sqrt[3]{\frac{12}{5x}}
Explanation:
Finding the inverse of a function means determining a new function that “undoes” the effect of the original. In other words, if f(a)=bf(a) = b, then f−1(b)=af^{-1}(b) = a. To do this, one swaps the roles of input and output and solves for the original variable.
The function given, f(x)=125x3f(x) = \frac{12}{5x^3}, is a rational function where xx is in the denominator and raised to a power. The process starts by renaming the function f(x)f(x) as yy to simplify notation. Then, to find the inverse, input and output are interchanged, and the resulting equation is solved for the new output variable yy.
After swapping xx and yy, algebraic manipulation is required to isolate yy. Multiplying both sides by 5y35y^3 clears the denominator. Then dividing by 5x5x and taking the cube root gives the expression for the inverse function.
This process works because the function is one-to-one and passes the horizontal line test, meaning each output corresponds to exactly one input. The inverse found, f−1(x)=125x3f^{-1}(x) = \sqrt[3]{\frac{12}{5x}}, will reverse the effect of the original function for all valid xx-values in its domain.