ind 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)=
To find the inverse of a function, you need to follow these steps:
Step 1: Replace the function notation with "y". This means that for each function, you'll replace "f(x)" with "y", "g(x)" with "y", and "h(x)" with "y".
a) f(x) = 6x + 10
Replace f(x) with y: y = 6x + 10
Step 2: Switch the roles of x and y. This means that you'll interchange x and y in the equation from Step 1.
a) Interchanging x and y: x = 6y + 10
Step 3: Solve the equation from Step 2 for y, which means isolating y on one side of the equation.
a) Solve for y: x - 10 = 6y
Divide by 6: (x - 10)/6 = y
Step 4: Replace y with the inverse notation "f^-1(x)".
a) Inverse of f(x): f^-1(x) = (x - 10)/6
Now let's move on to the other functions:
b) g(x) = 15x^2 - 8, with domain x ≥ 0
Replace g(x) with y: y = 15x^2 - 8
b) Interchanging x and y: x = 15y^2 - 8
b) Solve for y: 15y^2 = x + 8
Divide by 15: y^2 = (x + 8)/15
Take the square root: y = ±√((x + 8)/15)
Please note that we use the ± symbol here because the inverse of a parabola has two branches due to its symmetry. However, since the domain of g(x) is specified as x ≥ 0, we take the positive square root:
b) Inverse of g(x): g^-1(x) = √((x + 8)/15)
c) h(x) = 6x + 8
Replace h(x) with y: y = 6x + 8
c) Interchanging x and y: x = 6y + 8
c) Solve for y: x - 8 = 6y
Divide by 6: (x - 8)/6 = y
c) Inverse of h(x): h^-1(x) = (x - 8)/6
These are the inverse functions for each of the given functions.