Find the inverses of each of the functions below algebraically.
a.f(x)=3y+5x=18
b.h(t)=−4.9(t+3)^2+45.8
To find the inverse of a function algebraically, we need to follow a few steps:
Step 1: Replace the function's variable (usually denoted as "x") with "y".
Step 2: Swap the positions of "x" and "y".
Step 3: Solve the resulting equation for "y" to get the inverse function.
Step 4: Replace "y" with the traditional notation for inverse functions (usually denoted as "f^(-1)(x)").
Let's apply this process to the given functions:
a. f(x) = 3y + 5x = 18
Step 1: Replace "x" with "y".
3y + 5y = 18
Step 2: Swap the positions of "x" and "y".
3y + 5y = 18
Step 3: Solve the equation for "y".
8y = 18
y = 18/8
y = 9/4
Step 4: Replace "y" with "f^(-1)(x)" to get the inverse function.
f^(-1)(x) = 9/4
The inverse function of f(x) is f^(-1)(x) = 9/4.
b. h(t) = -4.9(t + 3)^2 + 45.8
Step 1: Replace "t" with "y".
h(y) = -4.9(y + 3)^2 + 45.8
Step 2: Swap the positions of "t" and "y".
y = -4.9(y + 3)^2 + 45.8
Step 3: Solve the equation for "y".
y = -4.9(y + 3)^2 + 45.8
y + 4.9(y + 3)^2 = 45.8
y + 4.9(y^2 + 6y + 9) = 45.8
y + 4.9y^2 + 29.4y + 44.1 = 45.8
4.9y^2 + 30.4y - 1.7 = 0
We can solve this quadratic equation to find the inverse function of h(t). However, note that the given function is not a one-to-one function, meaning it doesn't have an inverse over its entire domain. The inverse, if exists, would be a restricted function.