Show that f(x)=4x^5-8x^4 is one to one algebraically. Then find the inverse.
Not necessarily looking for the answer, i just don’t know how to put the problem together
Just looking at the graph, you can see that f(x) is not 1-1.
For example f(-0.559) = f(0.657) = f(1.984) = -1
So it has no single inverse.
For f to be 1-1, it must be monotonically increasing or decreasing. That means that f'(x) is always either positive or negative.
To show that a function is one-to-one algebraically, we need to prove that if f(a) = f(b), then a = b.
First, let's assume that f(a) = f(b), where a and b are two different values in the domain of the function f(x).
So, we have:
f(a) = f(b)
4a^5 - 8a^4 = 4b^5 - 8b^4
Now, our goal is to simplify this equation and show that a = b. Here's how:
4a^5 - 8a^4 = 4b^5 - 8b^4
Divide through by 4:
a^5 - 2a^4 = b^5 - 2b^4
Rearrange the equation:
a^5 - b^5 = 2a^4 - 2b^4
Factor:
(a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) = 2(a^4 - b^4)
Divide both sides by (a - b):
a^4 + a^3b + a^2b^2 + ab^3 + b^4 = 2(a^4 - b^4)
Simplify:
a^4 + a^3b + a^2b^2 + ab^3 + b^4 = 2a^4 - 2b^4
Rearrange:
-a^4 + a^3b + a^2b^2 + ab^3 - b^4 = 0
Factor:
-(a - b)(a^3 - a^2b + ab^2 - b^3) = 0
Since a and b are different values, a - b ≠ 0. Therefore, the only way for the equation to hold is if:
a^3 - a^2b + ab^2 - b^3 = 0
Now, let's find the inverse of the function f(x) = 4x^5 - 8x^4.
1. Start with f(x) = 4x^5 - 8x^4.
2. Replace f(x) with y: y = 4x^5 - 8x^4.
3. Swap the variables, so we have x in terms of y.
x = (y/4)^(1/5)
Therefore, the inverse function of f(x) is given by:
f^(-1)(x) = (x/4)^(1/5)
To show that a function is one-to-one algebraically, we need to prove that it has a unique inverse. A function has an inverse if it is both injective (one-to-one) and surjective (onto).
To demonstrate that f(x) = 4x^5 - 8x^4 is one-to-one, we can use a technique called the horizontal line test. It involves checking if any horizontal line intersects the graph of the function at more than one point.
1. Begin by assuming that f(a) = f(b), where a and b are two distinct values in the domain of the function.
2. Write out the equation using f(x):
f(a) = f(b)
4a^5 - 8a^4 = 4b^5 - 8b^4
3. Rearrange the equation to bring like terms together:
4a^5 - 4b^5 = 8a^4 - 8b^4
4. Factor out the common factors:
4(a^5 - b^5) = 8(a^4 - b^4)
5. Divide both sides of the equation by the common factor:
a^5 - b^5 = 2(a^4 - b^4)
6. Notice that a^5 - b^5 can be further factored using the difference of squares formula:
(a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) = 2(a^4 - b^4)
7. Divide both sides by (a - b):
a^4 + a^3b + a^2b^2 + ab^3 + b^4 = 2(a^4 - b^4)
8. Subtract 2(a^4 - b^4) from both sides:
a^4 + a^3b + a^2b^2 + ab^3 + b^4 - 2(a^4 - b^4) = 0
9. Simplify the equation further:
3a^4 + a^3b + a^2b^2 + ab^3 + 3b^4 = 0
10. This equation demonstrates that a^4 and b^4 are equal, as all other terms cancel out. Since a^4 = b^4, we can raise both sides to the 1/4 power:
(a^4)^(1/4) = (b^4)^(1/4)
11. Simplify to obtain:
|a| = |b|
From this simplified equation, we can conclude that a = b or a = -b. This shows that f(x) is one-to-one.
To find the inverse of f(x), follow these steps:
1. Replace f(x) with y:
y = 4x^5 - 8x^4
2. Swap x and y:
x = 4y^5 - 8y^4
3. Solve for y:
4y^5 - 8y^4 = x
4. Factor out a common factor:
4y^4(y - 2) = x
5. Divide both sides by 4(y - 2):
y^4 = x / (4(y - 2))
6. Take the fourth root of both sides:
y = (x / (4(y - 2)))^(1/4)
Thus, the inverse of f(x) is given by y = (x / (4(y - 2)))^(1/4).