Write a function of x that performs the following operations: Raise x to the ninth power, multiply by 5, and then add 6.
y=f(x)=
(b) Find the inverse to the function you found in part (a).
x=g(y)=
(a) f(x) = 5x^9 + 6
(b) g(y) = (y-6)/5
(a) The function y=f(x) is defined as follows:
y = (x^9) * 5 + 6
(b) To find the inverse function, we need to interchange the roles of x and y and solve for x. Let's start by rewriting the function equation:
x = (y^9) * 5 + 6
Now, let's solve for y:
x - 6 = (y^9) * 5
(x - 6)/5 = y^9
((x - 6)/5)^(1/9) = y
Therefore, the inverse function is:
x = g(y) = ((y - 6) / 5)^(1/9)
(a) To write a function that performs the operations of raising x to the ninth power, multiplying by 5, and then adding 6, we can follow these steps:
Step 1: Raise x to the ninth power by using the exponentiation operator (^). This can be done with the expression x^9.
Step 2: Multiply the result from step 1 by 5. We can multiply two numbers using the multiplication operator (*). So, we have 5 * (x^9).
Step 3: Add 6 to the result from step 2. We can use the addition operator (+) for this. The final function y = f(x) is given by:
y = f(x) = 5 * (x^9) + 6
(b) To find the inverse of the function y = f(x), we need to interchange the roles of x and y and solve for y.
Step 1: Start with the original function y = f(x) = 5*(x^9) + 6.
Step 2: Interchange x and y, so we have x = 5*(y^9) + 6.
Step 3: Now, solve for y. To do this, we need to isolate y on one side of the equation.
Subtract 6 from both sides of the equation: x - 6 = 5*(y^9).
Divide both sides of the equation by 5: (x - 6)/5 = y^9.
Step 4: To get the inverse function, take the ninth root of both sides of the equation. The ninth root of a number can be expressed using the exponent 1/9. So, we have:
((x - 6)/5)^(1/9) = y.
Therefore, the inverse function x = g(y) is given by:
x = g(y) = ((y - 6)/5)^(1/9).