Use the function f(x)=5x^3 − 4 to answer the questions.
A: What is f^−1 (x)?
B: What should be done to find the value of x that makes f(x)=74.125?
C: For what value of x does f(x)=74.125?
let's do B and C first
5x^3 − 4 = 74.125
5x^3 = 78.125
x^3 = 15.625
x = cuberoot(15.625) = 2.5
a) f(x)=5x^3 − 4 or
y = 5x^3 - 4
step1. Interchange the x and y variables
x = 5y^3 - 4
step2. Now solve this new equation for y
5y^3 = x + 4
y^3 = (x+4)/5
y = cuberoot [ (x+4)/5 ] or y = ( (x+4)/5 )^(1/3)
To answer these questions, we will be using the given function f(x) = 5x^3 - 4.
A: To find the inverse of a function f(x), denoted as f^−1(x), we need to follow these steps:
1. Replace f(x) with y. So, the original function becomes: y = 5x^3 - 4.
2. Swap the x and y variables, so the equation becomes: x = 5y^3 - 4.
3. Solve this equation for y. In this case, we need to find f^−1(x), so let's solve for y:
x = 5y^3 - 4
x + 4 = 5y^3
(x + 4) / 5 = y^3
∛((x + 4) / 5) = y
Therefore, f^−1(x) = (∛((x + 4) / 5)).
B: To find the value of x that makes f(x) = 74.125, we need to set up the equation:
f(x) = 5x^3 - 4 = 74.125
5x^3 = 78.125
Divide both sides by 5:
x^3 = 15.625
Take the cube root of both sides to solve for x:
x = ∛15.625
Therefore, to find the value of x, we take the cube root of 15.625, which is approximately 2.5.
C: For what value of x does f(x) = 74.125? We already set up the equation in the previous step:
f(x) = 5x^3 - 4 = 74.125
5x^3 = 78.125
Divide both sides by 5:
x^3 = 15.625
Take the cube root of both sides to solve for x:
x = ∛15.625
Therefore, the value of x that makes f(x) = 74.125 is approximately 2.5.