\(f(x)=\sqrt{5-x}+3\)
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. \(f(x)=\sqrt{5-x}+3\) changes to \(y=\sqrt{5-x}+3\)
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.
\(y=\sqrt{5-x}+3\) changes to \(x=\sqrt{5-y}+3\)
3. Solve for y
This step is the hardest. Transform the equation into the form of y=.
\(x=\sqrt{5-y}+3\) | Subtract 3 on both sides. |
\(x-3=\sqrt{5-y}\) | Square both sides to eliminate the square root. |
\((x-3)^2=\left(\sqrt{5-y}\right)^2\) | Expand the left hand side knowing that \((a-b)^2=a^2-2ab+b^2\). |
\(x^2-6x+9=5-y\) | Subtract 5 from both sides. |
\(-y=x^2-6x+4\) | Divide by -1. |
\(y=-x^2+6x-4\) | |
4. Consider Whether the Inverse is actually a Function
In this case, it is a function, so we are OK.
By definition, \(\sqrt{x^2}=|x|\). I know this because if you graph both functions, the output will be the same.
\(x^2+24=0\) | Subtract 24 from both sides. |
\(x^2=-24\) | Take the square root from both sides. |
\(|x|=\sqrt{-24}\) | The absolute value symbol means that the answer is in its positive and negative forms. |
\(x=\pm\sqrt{-24}\) | Now, let's change the square root to an imaginary form. We can apply the radical rule that \(\sqrt{-a}=\sqrt{-1}\sqrt{a}\) |
\(x=\pm\sqrt{24}\sqrt{-1}\) | We know that by definition, \(i=\sqrt{-1}\) |
\(x=\pm i\sqrt{24}\) | We can simplify the square root of 2 to its simplest radical form. |
\(x=\pm2i\sqrt{6}\) | |
In order to solve for x in this equation, we must perform a multitude of operations.
\(2x-6=\frac{3(10-2x)}{2}+\frac{3+5x}{2}\) | 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. |
\(x=9\) | |
This is indeed a messy problem. I'll solve for both variables, I guess.
1. Solve for a Variable
In this case, I will solve for x in the first equation, \(\sqrt{x^2+y^2}=100\).
\(\sqrt{x^2+y^2}=100\) | The first step is to square both sides so that we eliminate the square root symbol. |
\(x^2+y^2=10000\) | Subtract \(y^2\) from both sides. |
\(x^2=10000-y^2\) | Take the square root from both sides to isolate x. |
\(x=\sqrt{10000-y^2}\) | |
2. Use Substitution to eliminate the x and solve for y
Plug in \(\sqrt{10000-y^2}\) for x into the 2nd equation and solve for y. This is not going to look good...
\(\sqrt{\left(x+\frac{4800}{\sqrt{937}}\right)^2+\left(y+\frac{3800}{\sqrt{937}}\right)^2}=299.289496902\) | Plug in the value for x. |
\(\sqrt{\left(\sqrt{10000-y^2}+\frac{4800}{\sqrt{937}}\right)^2+\left(y+\frac{3800}{\sqrt{937}}\right)^2}=299.289496902\) | Square both sides to eliminate the square root. |
\(\textcolor{blue}{\left(\sqrt{10000-y^2}+\frac{4800}{\sqrt{937}}\right)^2}+\left(y+\frac{3800}{\sqrt{937}}\right)^2=299.289496902^2\) | In both binomials, we must follow the rule that \((a+b)^2=a^2+2ab+b^2\). I'll do the first binomial first, in blue. |
\(\left(\sqrt{10000-y^2}+\frac{4800}{\sqrt{937}}\right)^2=\left(\sqrt{10000-y^2}\right)^2+2\sqrt{10000-y^2}*\frac{4800}{\sqrt{937}}+\left(\frac{4800}{\sqrt{937}}\right)^2\) | Simplify this. Here we go... |
\(10000-y^2+\frac{9600\sqrt{10000-y^2}}{\sqrt{937}}+\frac{23040000}{937}\) | Now, let's remember that this equals the bit in blue. Now, let's expand the other part, in normal text.This time, I'll expand it without showing much. |
\(\left(y+\frac{3800}{\sqrt{937}}\right)^2\) | Expand this using the same technique as above. |
\(y^2+\left(2y*\frac{3800}{\sqrt{937}}\right)+\left(\frac{3800}{\sqrt{937}}\right)^2\) | Simplify further. |
\(y^2+\frac{7600y}{\sqrt{937}}+\frac{14440000}{937}\) | |
Time to do the simplification process.
\(10000-y^2+\frac{9600\sqrt{10000-y^2}}{\sqrt{937}}+\frac{23040000}{937}+y^2+\frac{7600y}{\sqrt{937}}+\frac{14440000}{937}\) | To make this easier to digest, I will rearrange the terms. |
\(y^2-y^2+\frac{9600\sqrt{10000-y^2}}{\sqrt{937}}+\frac{7600y}{\sqrt{937}}+10000+\frac{23040000}{937}+\frac{14440000}{937}\) | Now, let's do the simplification process. |
\(\frac{9600\sqrt{10000-y^2}+7600y}{\sqrt{937}}+50000\) | Great! Now that we have simplified as much as possible, let's reinsert this into the equation. |
\(\frac{9600\sqrt{10000-y^2}+7600y}{\sqrt{937}}+50000=299.289496902^2\) | We have to get rid of the remaining fraction by multiplying by the square root of 937 on all sides. |
\(9600\sqrt{10000-y^2}+7600y+50000\sqrt{937}=299.289496902^2\sqrt{937}\) | Subtract \(7600y+50000\sqrt{937}\) |
\(9600\sqrt{10000-y^2}=229.289496902^2\sqrt{937}-7600y-50000\sqrt{937})\) | Doing some more simplifying... |
You know what...
This is really boring and tedious...
There are 2 ordered pair solutions to this. They are the following:
\(x≈68.53740254003346, y≈72.81912147963209\)
\(x≈86.60254037943613, y≈49.99999999828134\)
.
To find the distance of two numbers on the number line, simply subtract the 2 numbers.
\(-2\frac{1}{2}-(-5\frac{3}{4})\) | Subtracting a negative is equivalent to adding a positive. |
\(-2\frac{1}{2}+5\frac{3}{4}\) | Now convert each fraction to an improper fraction. Doing this is the first step to allow us to calculate the distance. |
\(-2\frac{1}{2}=-\frac{2*2+1}{2}=\frac{-5}{2}\) | Now, convert the other fraction to an improper fraction. |
\(5\frac{3}{4}=\frac{4*5+3}{4}=\frac{23}{4}\) | Now, reinsert these improper fraction back into the expression. |
\(\frac{-5}{2}+\frac{23}{4}\) | In order to add these fractions, we must create a common denominator. Multiply the first fraction by 2/2 to achieve this. |
\(\frac{-10}{4}+\frac{23}{4}\) | Now, add the fractions together. |
\(\frac{13}{4}=3.25\) | This is the distance in between the two numbers on the number line. |