Given f(x)=4x＾2+6x-4 if g(x-2)=f(x), then g(x)=?

Question

Given f(x)=4x＾2+6x-4

if g(x-2)=f(x), then g(x)=?

Answer 1

g(x - 2) = 4x² + 6x - 4 so our goal is to write an expression where the variable is "x - 2" instead of "x" for the expression 4x² + 6x - 4. That will then tell us what g(x) is.

Start with replacing the "x" in 4x² with "x - 2" so 4(x - 2)² which is equal to 4(x² - 4x + 4) or 4x² - 16x + 16. To get back to 4x² + 6x - 4 we need to add in "22x - 20" So, we can now write f(x) as 4(x - 2)² + 22x - 20. Now, we want to replace the second "x" (the one in the "22x" with "x - 2" so: 4(x - 2)² + 22(x - 2). This multiplies out to be 4x² - 16x + 16 + 22x - 44 or 4x² + 6x - 28. To get back to our original f(x), we need to add 24. So our f(x) is now 4(x - 2)² + 22(x - 2) + 24. I encourage you to multiply this out or graph it to ensure it is the same as our original f(x).

Written as f(x) = 4(x - 2)² + 22(x - 2) + 24, it is easy to see that g(x) = 4x² + 22x + 24 because g(x - 2) = f(x)

Answer 2

This type of questions are tricky. If you would know g(x) formula, then getting g(x-2) is easy - you just need to replace x with x-2 in its formula.

But here you need to work backwards, which is always more complicated and error prone.

So

g(x-2) = 4x²+6x-4

To get g(x) you need actually to replace X with (X+2) - that is the trick.

So

g(x) = 4(x+2)²+6(x+2) - 4

Opening parenthesis you will get

4x²+22x+24

You can verify this answer by replacing x with (x-2) - you will get 4x²+6x-4