Given f(x)=7-2(x-1)^2, x>= 1, determine f inverse (5).
Well, let's start by finding f inverse (x). To do that, we need to swap the x and y variables and solve for y.
So, let's rewrite the equation f(x)=7-2(x-1)^2 as x=7-2(y-1)^2.
Now, let's solve for y. Subtract 7 from both sides: -x+7=-2(y-1)^2.
Divide by -2: (x-7)/-2=(y-1)^2.
Take the square root: sqrt((x-7)/-2)=y-1.
Add 1 to both sides: sqrt((x-7)/-2) + 1 = y.
So, the inverse function of f(x) is f inverse (x) = sqrt((x-7)/-2) + 1.
Now, to answer the actual question, we need to find f inverse (5).
Plugging in x = 5 into the equation f inverse (x) = sqrt((x-7)/-2) + 1, we get:
f inverse (5) = sqrt((5-7)/-2) + 1
f inverse (5) = sqrt((-2)/-2) + 1
f inverse (5) = sqrt(1) + 1
f inverse (5) = 1 + 1
f inverse (5) = 2
So, f inverse (5) = 2.
To find the inverse of a function, we need to solve for x in terms of y. In this case, we want to find f^(-1)(5), which means finding the value of x when f(x) = 5.
First, let's start with the original function f(x) = 7 - 2(x - 1)^2.
Step 1: Replace f(x) with 5:
5 = 7 - 2(x - 1)^2
Step 2: Subtract 7 from both sides:
5 - 7 = -2(x - 1)^2
Step 3: Simplify:
-2 = -2(x - 1)^2
Step 4: Divide both sides by -2:
1 = (x - 1)^2
Step 5: Take the square root of both sides:
±√1 = ±√(x - 1)^2
Simplifying:
±1 = ±(x - 1)
Step 6: Solve for x:
x - 1 = 1 or x - 1 = -1
For the first equation:
x - 1 = 1
Adding 1 to both sides:
x = 2
For the second equation:
x - 1 = -1
Adding 1 to both sides:
x = 0
So, the solution is x = 2 or x = 0.
Therefore, the inverse of the function f(x) = 7 - 2(x - 1)^2 is f^(-1)(5) = 2 or f^(-1)(5) = 0.
To find the inverse of a function, we need to switch the roles of x and y and solve for y.
Given function: f(x) = 7 - 2(x - 1)^2
Step 1: Replace f(x) with y.
y = 7 - 2(x - 1)^2
Step 2: Switch x and y in the equation.
x = 7 - 2(y - 1)^2
Step 3: Solve the equation for y.
x = 7 - 2(y - 1)^2
x - 7 = -2(y - 1)^2
(x - 7)/(-2) = (y - 1)^2
sqrt((x - 7)/(-2)) = y - 1
sqrt((x - 7)/(-2)) + 1 = y
Here, we have expressed y in terms of x, so y is the inverse function of f(x).
The inverse function of f(x) is f^(-1)(x) = sqrt((x - 7)/(-2)) + 1.
To find f^(-1)(5), substitute x = 5 into the inverse function.
f^(-1)(5) = sqrt((5 - 7)/(-2)) + 1
= sqrt((-2)/(-2)) + 1
= sqrt(1) + 1
= 1 + 1
= 2
Therefore, f^(-1)(5) = 2.
original: y = 7 - 2(x-1)^2
inverse : x = 7 - 2(y-1)^2
2(y-1)^2 = 7 - x
(y-1)^2 = (7-x)/2
y-1 = ± √[(7-x)/2]
y = 1 ± √[(7-x)/2] but x ≥ 1 in the original, which means only one wing of the parabola.
When we sketch that wing and reflect it in the line
y = x we see that our inverse must be
y = 1 + √[(7-x)/2]
check: let x=3 in original, y = 7-8 = -1
let x = -1 in inverse , y = 1 + √4 = 3
let x=10, y = 7- -155
in inverse, let x = -155
y = 1 + √81 = 10