For the function f(x)=(8-2x)^2, find f^-1. Determine weather f^-1 is a function.
To find the inverse of a function, we need to switch the roles of x and y and solve for y.
Step 1: Replace f(x) with y:
y = (8 - 2x)^2
Step 2: Swap x and y:
x = (8 - 2y)^2
Step 3: Solve for y:
Take the square root of both sides to undo the squaring operation.
√x = 8 - 2y
Step 4: Isolate y:
Rearrange the equation to isolate y.
2y = 8 - √x
y = (8 - √x)/2
So, the inverse of the function f(x) = (8 - 2x)^2 is f^(-1)(x) = (8 - √x)/2.
To determine whether f^(-1)(x) is a function, we need to check if each input corresponds to a unique output. In other words, every value of x should produce exactly one value of y.
In this case, the expression f^(-1)(x) = (8 - √x)/2 involves taking the square root of x. For the inverse to be a function, the expression under the square root (√x) cannot be negative since square roots of negative numbers are not defined in the real number system.
Therefore, f^(-1)(x) is a function only if x ≥ 0.
To find the inverse of a function f(x), you need to follow these steps:
Step 1: Replace f(x) with y. The equation becomes y = (8-2x)^2.
Step 2: Swap the x and y variables. The equation becomes x = (8-2y)^2.
Step 3: Solve the new equation for y. Begin by taking the square root of both sides to eliminate the square. This gives us √x = 8-2y.
Step 4: Isolate y by moving the term with y to one side. Subtract 8 from both sides to get √x - 8 = -2y.
Step 5: Divide both sides by -2 to solve for y. This gives us y = (8 - √x) / 2.
So, the inverse function f^(-1)(x) is given by f^(-1)(x) = (8 - √x) / 2.
To determine whether f^(-1) is a function, we need to check whether it passes the horizontal line test.
The horizontal line test states that if any horizontal line intersects the graph of a function at more than one point, then the inverse is not a function.
In this case, since the inverse function is a linear function (a straight line), it passes the horizontal line test. Therefore, f^(-1) is indeed a function.
To get the inverse, what you'll do is:
(1) let f(x)=y, and replace x with y and y with x:
f(x) = (8 - 2x)^2
y = (8 - 2x)^2
x = (8 - 2y)^2
(2) solve for y, and substitute back the y=f(x):
x = (8 - 2y)^2
sqrt(x) = 8 - 2y
sqrt(x) - 8 = -2y
y = (sqrt(x) - 8) / (-2)
y = (8 - sqrt(x)) / 2
f(x) = (8 - sqrt(x)) / 2
hope this helps~ `u`