Find the inverse for each of the following functions.
a) f(x)=6x+10
f−1(y)=
b) g(x)=15x2−8, with domain x≥0
g−1(y)=
c) h(x)=6x+8
h−1(y)=
f = 6x+10
f-10 = 6x
(f-10)/6 = x = f^-1
g = 15x^2-8
g+8 = 15x^2
(g+8)/15 = x^2
√((g+8)/15) = x = g^-1
I'm sure you can handle the last one, right?
To find the inverse of a function, we can follow these steps:
Step 1: Replace the function notation f(x) with y.
Step 2: Swap the roles of x and y, which means interchange x and y in the equation.
Step 3: Solve the new equation for y to obtain the inverse function in terms of y.
Let's go through each function and find their inverses:
a) f(x) = 6x + 10
Step 1: Replace f(x) with y:
y = 6x + 10
Step 2: Swap x and y:
x = 6y + 10
Step 3: Solve for y:
x - 10 = 6y
Divide both sides by 6:
(y-10)/6 = y/6
Simplify:
y/6 - 10/6 = y/6
Combine like terms:
y/6 - 5/3 = y/6
Multiply through by 6 to eliminate the denominators:
6(y/6) - 6(5/3) = 6(y/6)
simplifying, we get:
y - 10 = y
Now isolate y on one side:
y - y = 10
0 = 10
This is a contradiction, which means the function does not have an inverse. Therefore, there is no inverse function for f(x) = 6x + 10.
b) g(x) = 15x^2 - 8, with domain x ≥ 0
Step 1: Replace g(x) with y:
y = 15x^2 - 8
Step 2: Swap x and y:
x = 15y^2 - 8
Step 3: Solve for y:
x + 8 = 15y^2
Divide both sides by 15:
(x + 8)/15 = y^2
Taking the square root of both sides (with consideration to the positive square root due to the domain):
√((x + 8)/15) = y
Now, we have the inverse function g^(-1)(y) = √((x + 8)/15)
c) h(x) = 6x + 8
Step 1: Replace h(x) with y:
y = 6x + 8
Step 2: Swap x and y:
x = 6y + 8
Step 3: Solve for y:
x - 8 = 6y
Divide both sides by 6:
(y-8)/6 = y/6
Simplify:
y/6 - 8/6 = y/6
Combine like terms:
y/6 - 4/3 = y/6
Multiply through by 6 to eliminate the denominators:
6(y/6) - 6(4/3) = 6(y/6)
Simplifying, we get:
y - 8 = y
Now isolate y on one side:
y - y = 8
0 = 8
This is a contradiction, which means the function does not have an inverse. Therefore, there is no inverse function for h(x) = 6x + 8.