Question 160663This question is from textbook Algebra and Trigonometry Structure and Method book 2
: I have been struggling on how to figure this problem out and I was wondering if someone could help me? Please and Thank You!! I would deeply appreciate it!!
Let f(x)=x^2-1 and g(x)=1-2x. Find the indicated values.
f(a+1)-f(a)
This question is from textbook Algebra and Trigonometry Structure and Method book 2
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! f(x) = x^2 - 1
g(x) = 1-2*x
f(a+1) - f(a)
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i had such a good time on the last one that i decided i would do something for this one as well.
this problem is different in that g(x) doesn't come into the picture.
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they are telling you that f(x) = x^2 - 1 which you can translate as y = f(x) = x^2 - 1 which means y is a function of x and the function is x^2 - 1.
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it's just a different way of looking at it that may make it a little clearer.
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this problem states that f(x) = x^2 - 1
you know what the function does and all you need to do is substitute a requested value for x.
if you substitute 3 for x, then the function becomes f(3) = 3^2 - 1.
if you substitute 342 for x, then the function becomes (f(342) = (342)^2 - 1
in this case, they are asking for f(a+1) - f(a).
this means they are asking you to substitute (a+1) for x in the first part of the equation and they are asking you to substitute (a) for x in the second part of the equation.
first solve each one individually.
since f(x) = x^2 - 1, then f(a) must equal a^2 - 1.
since f(x) = x^2 - 1, then f(a+1) must equal (a+1)^2 - 1
you now have the piece parts and can solve the whole thing.
f(a+1) - f(a) becomes ((a+1)^2 - 1) - (a^2 - 1).
(a+1)^2 is the same as (a+1)*(a+1) which equals a^2 + 2a + 1.
so the equation becomes (a^2 + 2*a + 1 - 1) - (a^2 - 1)
you can just remove the parentheses from the left side since that part doesn't have any operation being performed on it.
in order to remove the parentheses from the right side, you have to take into account the operation being performed on it, which is - for subtraction. you have to subtract each element within the parentheses separately and you have to remember that when you minus a plus you get a minus and when you minus a minus you get a plus. so - (+a^2) gets you -a^2, and - (-1) gets you +1
after you remove the parentheses, the equation becomes
a^2 + 2*a + 1 - 1 - a^2 + 1 which becomes 2*a + 1 because +a^2 and -a^2 cancel out, and +1 and -1 cancel out leaving only 2*a and + 1.
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answer is:
f(a+1) - f(a) = 2*a + 1.
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