f(x)=√5−x+3
Changing this to its inverse requires a few steps.
1. Change f(x) to y.
This step is pretty simple. We are switching from function notation to y=-notation. f(x)=√5−x+3 changes to y=√5−x+3
2. Interchange x and y
This step is also quite simple; replace all instances of x with y and all instance of y with x.
y=√5−x+3 changes to x=√5−y+3
3. Solve for y
This step is the hardest. Transform the equation into the form of y=.
x=√5−y+3 | Subtract 3 on both sides. |
x−3=√5−y | Square both sides to eliminate the square root. |
(x−3)2=(√5−y)2 | Expand the left hand side knowing that (a−b)2=a2−2ab+b2. |
x2−6x+9=5−y | Subtract 5 from both sides. |
−y=x2−6x+4 | Divide by -1. |
y=−x2+6x−4 | |
4. Consider Whether the Inverse is actually a Function
In this case, it is a function, so we are OK.
By definition, √x2=|x|. I know this because if you graph both functions, the output will be the same.
x2+24=0 | Subtract 24 from both sides. |
x2=−24 | Take the square root from both sides. |
|x|=√−24 | The absolute value symbol means that the answer is in its positive and negative forms. |
x=±√−24 | Now, let's change the square root to an imaginary form. We can apply the radical rule that √−a=√−1√a |
x=±√24√−1 | We know that by definition, i=√−1 |
x=±i√24 | We can simplify the square root of 2 to its simplest radical form. |
x=±2i√6 | |
In order to solve for x in this equation, we must perform a multitude of operations.
2x−6=3(10−2x)2+3+5x2 | Multiply by 2 on both sides to eliminate the pesky fractions. |
4x−12=3(10−2x)+3+5x | Distribute the 3 into the term 10-2x. |
4x−12=30−6x+3+5x | Simplify the right hand side by combining the like terms. |
4x−12=−x+33 | Add x to both sides to get the x's on one side of the equation. |
5x−12=33 | Add 12 to both sides. |
5x=45 | Divide by 5 on both sides of the equation. |
x=9 | |