Find the inverse function...

Question

Find the inverse function for f(x)=√2x-6

Answer 1

Hi Alyce,

it's tough to know exactly what the problem was originally, what with text formatting here. Let's assume that it was (using parentheses to assure exact meanings)

f(x) = (sqrt(2))*x - 6

then let f(x) = y (for convenience) and solve for x = :

y = sqrt(2) * x - 6

y + 6 = sqrt(2) * x

x = (y + 6) / sqrt(2)

to make this into the standard format of a function, switch the variable terminology:

y = f(x) = (x + 6) / sqrt(2)

If that wasn't the exact problem you intended, you still go through the same steps in solving it, that is, set y as your temporary, dummy variable, and then solve for "x" in the original equation. You might have to use the squaring operation (to get rid of a square root) and so on. Make sure that, after you get to a final answer, that you also state any limits on variable domain -- i.e., so that you don't end up taking the square root of a negative value, in "either direction" on the function and inverse function set.

To cast this in more general terms: given your current function, and using the rules of PEMDAS, "peel away" successively all the operations done on "x", by using their respective inverse operations on the expressions on both sides of the equals sign. When you get down to just "x =" then switch the variables around, since you always want to state your answer function as f(x) = some expression using "x".

Answer 2

f(x) = (square root of 2)*x - 6

To find the inverse of f(x), replace the x's and f(x) by each other (interchange them).

x = (square root of 2)* f(x) - 6

x + 6 = (square root of 2) * f(x)

f(x) = (x + 6) / (square root of 2)

now rationalize the denominator by multiplying by (square root of two) / (square root of two)

f(x) = (x + 6) * (square root of 2) / 2

f(x) = (1/2)(square root of 2)(x + 6)

This is actually the inverse of f(x), which is written as f^-1 (x)

f^-1 (x) = (1/2)(square root of 2)(x + 6)

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