which of the following represents f(x)?

Question

If f(x-3)=x²+x+1, which of the following represents f(x)?

Answer 1

f(x - 3) means that f(x) is translated 3 units to the right.  Therefore,

 

f(x) = (x + 3)² + (x + 3) + 1

f(x) = (x² + 6x + 9) + x + 3 + 1

f(x) = x² + 7x + 13

 

 

Now to check this, we evaluate f(x) when x=x-3.  We should get back the original problem.

 

f(x - 3) = (x - 3)² + 7(x - 3) + 13

f(x - 3) = (x² - 6x + 9) + 7x - 21 + 13

f(x - 3) = x² + x + 1

 

 

This checks out.

Answer 2

f(x - 3) is f(x) shifted three units to the right.

Thersfore to determine f(x), f(x - 3) must be shifted three unite to the left

f(x) = (x + 3)² + (x + 3) + 1

 

 

Answer 3

The graph of y = f(x-3) is obtained from the graph of y = f(x) by translating the graph of y = f(x) three units to the right.  

 

So, to obtain the graph of y = f(x) from the graph of y = f(x-3), translate the graph of y = f(x-3) three units to the left.  That is, in the formula for f(x-3), replace x by x+3.

 

So, f(x) = (x+3)²+(x+3)+1

 

           = x²+6x+9+x+3+1 = x²+7x+13