Let f(x)=(x-2)^3+8
a. Show that this function is one-to-one algebraically
b, Find the inverse of f(x)
a. To show that the function f(x) is one-to-one algebraically, we need to show that for any two distinct values of x, their corresponding y-values will also be distinct.
Let's assume two different values of x, x1 and x2, such that x1 ≠ x2.
f(x1) = (x1 - 2)^3 + 8
f(x2) = (x2 - 2)^3 + 8
Now, we will show that if f(x1) = f(x2), then x1 = x2:
If f(x1) = f(x2), then (x1 - 2)^3 + 8 = (x2 - 2)^3 + 8
Subtracting 8 from both sides,
(x1 - 2)^3 = (x2 - 2)^3
Taking the cube root of both sides,
x1 - 2 = x2 - 2
Cancelling out "2" from both sides,
x1 = x2
Hence, we have shown that for any two distinct values of x, their corresponding y-values will also be distinct. Therefore, the function f(x) is one-to-one algebraically.
b. To find the inverse of f(x), we will exchange the roles of x and y and solve for y.
Let y = (x - 2)^3 + 8
To find the inverse, we need to solve this equation for x.
Step 1: Swap x and y.
x = (y - 2)^3 + 8
Step 2: Solve for y.
x - 8 = (y - 2)^3
Taking the cube root of both sides,
∛(x - 8) = y - 2
Adding 2 to both sides,
y = ∛(x - 8) + 2
Therefore, the inverse of f(x) is given by:
f^(-1)(x) = ∛(x - 8) + 2
a. To show that the function f(x) = (x - 2)^3 + 8 is one-to-one algebraically, we need to demonstrate that each input x corresponds to a unique output y.
Let's assume that there are two different inputs, x1 and x2, such that f(x1) = f(x2).
Substituting x1 and x2 into the equation, we have:
(x1 - 2)^3 + 8 = (x2 - 2)^3 + 8
If we expand both sides, we get:
x1^3 - 6x1^2 + 12x1 - 8 = x2^3 - 6x2^2 + 12x2 - 8
Rearranging the terms, we have:
x1^3 - 6x1^2 + 12x1 = x2^3 - 6x2^2 + 12x2
Now, we assume that x1 is not equal to x2. Without loss of generality, let's assume x1 > x2. Then, since the cubic function is an increasing function, we have:
x1^3 > x2^3 [because x1 > x2]
Similarly, since x1 > x2, we have:
-6x1^2 + 12x1 > -6x2^2 + 12x2
Therefore, we can conclude that if x1 ≠ x2, then f(x1) ≠ f(x2). This means that the function f(x) = (x - 2)^3 + 8 is one-to-one algebraically.
b. To find the inverse of f(x), we will swap x and y and then solve for y.
Let's start with f(x) = (x - 2)^3 + 8.
Swap x and y:
x = (y - 2)^3 + 8
Now, solve this equation for y.
Subtract 8 from both sides:
x - 8 = (y - 2)^3
Take the cube root of both sides:
∛(x - 8) = y - 2
Add 2 to both sides:
y = ∛(x - 8) + 2
So, the inverse of f(x) is given by:
f^(-1)(x) = ∛(x - 8) + 2
To show that a function is one-to-one algebraically, we need to prove that if f(a) = f(b), then a = b.
a. Let's assume f(a) = f(b) and prove that a = b.
Given: f(a) = f(b)
f(a) = (a-2)^3 + 8
f(b) = (b-2)^3 + 8
Since f(a) = f(b), we can set them equal to each other:
(a-2)^3 + 8 = (b-2)^3 + 8
By subtracting 8 from both sides, we get:
(a-2)^3 = (b-2)^3
Take the cube root of both sides:
∛[(a-2)^3] = ∛[(b-2)^3]
Simplifying further, we have:
a-2 = b-2
Adding 2 to both sides, we find:
a = b
Since we have shown that if f(a) = f(b), then a = b, f(x) is one-to-one algebraically.
b. To find the inverse of f(x), we need to swap the roles of x and y and solve for y.
Step 1: Replace f(x) with y:
y = (x-2)^3 + 8
Step 2: Swap x and y:
x = (y-2)^3 + 8
Step 3: Solve for y:
x - 8 = (y-2)^3
∛(x - 8) = y - 2
Step 4: Add 2 to both sides:
∛(x - 8) + 2 = y
Thus, the inverse of f(x) is given by:
f^(-1)(x) = ∛(x - 8) + 2