Find inverse of f if f(x)= x^2-4x+3, (for x is smaller than and equal to 2).

First prove that f(x) is one to one in the defined domain of f and then obtain the inverse function.
I know how to find the inverse. We just switch x and y.
so y=x^2-4x+3 becomes
x=y^2-4y+3
Inverse of f= y^2-4y+3
How do I prove that it is one to one?
I know we prove by making a= b but I'm very confused with this problem

f(x) = (x-1)(x-3) So, the vertex is at x=2. The restriction on the domain ensures that we only have the left branch of the parabola, so f is 1-to-1 and will have an inverse.

x = y^2-4y+3
x = (y-2)^2 - 1
x+1 = (y-2)^2
y-2 = ±√(x+1)
we only want the left branch, so

y-2 = -√(x+1)
y = 2-√(x+1)

Note that the range of the inverse is y<=2, the domain of the original function. The domain of the inverse is x >= -1, which is the range of the original f(x).

To prove that a function is one-to-one, we need to show that for any two different inputs, the outputs are also different.

Let's assume we have two values of x, say a and b, such that a ≠ b. We need to show that f(a) ≠ f(b).

So, let's plug in a and b into the function f(x) = x^2 - 4x + 3:

f(a) = a^2 - 4a + 3
f(b) = b^2 - 4b + 3

To prove that f(a) ≠ f(b), we can start by assuming that f(a) = f(b) and then show that it leads to a contradiction.

Assume: f(a) = f(b)
Then: a^2 - 4a + 3 = b^2 - 4b + 3

Now, subtracting 3 from both sides:
a^2 - 4a = b^2 - 4b

Rearranging the terms:
a^2 - b^2 - 4a + 4b = 0

Factoring the difference of squares (a^2 - b^2):
(a - b)(a + b) - 4(a - b) = 0

Now, we can factor out (a - b) from both terms:
(a - b)(a + b - 4) = 0

To prove that f(a) ≠ f(b), we need to show that (a - b)(a + b - 4) ≠ 0.

Since we know a ≠ b (given in the question) and the second factor is a constant (a + b - 4), the only way the product can be zero is if (a - b) = 0. But, a - b = 0 implies that a = b, which contradicts our initial assumption that a ≠ b.

Thus, we have shown that f(a) ≠ f(b) for any a ≠ b. Hence, the function f(x) = x^2 - 4x + 3 is one-to-one in the domain where x is smaller than or equal to 2.

Now, to find the inverse function, we can switch x and y in the equation:
x = y^2 - 4y + 3

Let's solve for y:

x = y^2 - 4y + 3

Rearranging the terms:
y^2 - 4y = x - 3

Completing the square:
y^2 - 4y + 4 = x - 3 + 4
(y - 2)^2 = x + 1

Taking the square root of both sides:
y - 2 = ±√(x + 1)

Adding 2 to both sides:
y = 2 ± √(x + 1)

Now, the inverse function of f(x) = x^2 - 4x + 3, where x is smaller than or equal to 2, is given by:
f^(-1)(x) = 2 ± √(x + 1)

To prove that a function is one-to-one, we need to show that if any two values in the domain of the function have the same image, then the two values must be the same. In other words, if f(a) = f(b), then a = b.

In this case, we are given the function f(x) = x^2 - 4x + 3, for x ≤ 2. To prove that f(x) is one-to-one in this domain, we can use a direct proof.

Step 1: Assume that f(a) = f(b), where a and b are two arbitrary values in the domain of f.

f(a) = f(b) translates to (a^2 - 4a + 3) = (b^2 - 4b + 3).

Step 2: We need to show that a = b.

To solve this equation, we can rearrange it to obtain a quadratic equation and then find its roots:

a^2 - 4a + 3 = b^2 - 4b + 3
a^2 - b^2 - 4a + 4b = 0
(a - b)(a + b) - 4(a - b) = 0
(a - b)(a + b - 4) = 0

Now, we have two possible cases:
1. (a - b) = 0: This implies a = b.
2. (a + b - 4) = 0: This implies a + b = 4.

For the first case, a = b, which directly proves that the function is one-to-one.

For the second case, a + b = 4, we need to check if such values violate the given domain, which is x ≤ 2. Since a and b are in the domain, their sum should not exceed 2. However, a + b = 4 violates this condition, indicating that it is not a valid solution.

Since we have shown that there are no valid solutions for a + b = 4, we can conclude that the only solution for f(a) = f(b) is a = b. Therefore, the function f(x) = x^2 - 4x + 3, for x ≤ 2, is one-to-one in its domain.

Now that we have established the one-to-one property, we can proceed to find the inverse function by swapping x and y in the original equation:

x = y^2 - 4y + 3

To solve for y, we rearrange the equation and solve the resulting quadratic equation for y:

y^2 - 4y + (3 - x) = 0

By solving this quadratic equation, we can obtain the inverse function of f(x).