find the inverse of each function. is the inverse a function? 1) f(x)=-2x+9 2) f(x) x^2+2
its -2x+9
To find the inverse of a function, we need to switch the roles of x and y and solve for y. If the resulting equation can be expressed as y = some function of x, then the inverse is a function.
1) Let's find the inverse of the function f(x) = -2x + 9:
First, we switch the roles of x and y:
x = -2y + 9
Next, we solve for y:
2y = -x + 9
Divide both sides by 2:
y = (-1/2)x + 9/2
The inverse function of f(x) = -2x + 9 is given by:
f^(-1)(x) = (-1/2)x + 9/2
Since the inverse function f^(-1)(x) can be expressed as a function of x, it is a function.
2) Let's find the inverse of the function f(x) = x^2 + 2:
Again, we switch the roles of x and y:
x = y^2 + 2
Now, we solve for y:
y^2 = x - 2
Taking the square root of both sides, we have:
y = ±√(x - 2)
Since the inverse function has a ± sign, it is not a function. This is because for the same input, we have two possible outputs, making it violate the definition of a function.
Therefore, the inverse of the function f(x) = x^2 + 2 is not a function.
An easy way to find the inverse is
y = 2x+ 9 switch y and x
x = 2y +9 solve for y
(x-9)/2 = y
f^-1(x) (notation for inverse)
f^-1(x) =(x-9)/2
y = x^2 + 2
x = y^2 + 2
square root of (x-2) = y
f^-1(x) = square root of (x-2)