Let p(x)=2x^3-113 and let q be the inverse of p. Find q(137). Please explain in steps. Thank you.
To find q(137), we need to first find the inverse of p(x) and then substitute 137 into it. Let's go step by step to find q(137):
Step 1: Find the inverse of p(x)
To find the inverse, we need to swap the x and y variables and solve for y. So, we have:
x = 2y^3 - 113
Step 2: Solve for y
Rearranging the equation to solve for y:
2y^3 = x + 113
y^3 = (x + 113) / 2
y = ((x + 113) / 2)^(1/3)
Step 3: Substitute 137 into the inverse function
Now, replace x with 137 in the inverse function to find q(137):
q(137) = ((137 + 113) / 2)^(1/3)
Step 4: Simplify the expression
Evaluating the expression within the parentheses first:
q(137) = (250 / 2)^(1/3)
q(137) = 125^(1/3)
Step 5: Calculate the cube root of 125
Taking the cube root of 125:
q(137) = 5
Answer: q(137) = 5
So, the value of q(137) is 5.