Given f(x)=3x^2+4, evaluate each of the following:
a.) f(x+2)
b.) f(x+8)-f(x)/h, h doesn’t =0
f(x)=3x^2+4
f(x+2) = 3(x+2)^2 + 4
= 3(x^2 + 4x + 4) + 4
= 3x^2 + 12x + 12 + 4
= 3x^2 + 12x + 16
f(x+h)-f(x)/h <----- I am sure you have a typo and you meant h instead of 8, since I know where this is leading to
= ( (3(x+h)^2 + 4 - (3x^2 + 4) )/h
= (3x^2 + 6xh + 3h^2 + 4 - 3x^2 - 4)/h
= (6xh + 3h^2)/h
= 6x + 3h , h ≠ 0
=
Yes, it is 6x + 3h
in your next section of your topic you will find the word "limit" in front of
each expression of the solution, I bet you can't wait to get there.
Thank you does that mean the final answer for part b is 6x+3h?
To evaluate the given expressions, we need to substitute the given value into the function and simplify the expression. Let's go through each part:
a.) To evaluate f(x+2), we need to substitute (x+2) into the function f(x) = 3x^2 + 4:
f(x+2) = 3(x+2)^2 + 4
To simplify this expression, we first expand the square:
f(x+2) = 3(x^2 + 4x + 4) + 4
Next, distribute the 3:
f(x+2) = 3x^2 + 12x + 12 + 4
Finally, combine like terms to get the simplified expression:
f(x+2) = 3x^2 + 12x + 16
Therefore, the expression f(x+2) simplifies to 3x^2 + 12x + 16.
b.) To evaluate f(x+8) - f(x) / h (where h ≠ 0), we need to calculate the values separately and then divide:
First, let's calculate f(x+8):
f(x+8) = 3(x+8)^2 + 4
Expanding the square:
f(x+8) = 3(x^2 + 16x + 64) + 4
Distributing the 3:
f(x+8) = 3x^2 + 48x + 192 + 4
Combining like terms:
f(x+8) = 3x^2 + 48x + 196
Next, let's calculate f(x):
f(x) = 3x^2 + 4
Now we can substitute these values into the expression f(x+8) - f(x) / h:
[f(x+8) - f(x)] / h = [3x^2 + 48x + 196 - (3x^2 + 4)] / h
Simplifying the numerator:
[3x^2 + 48x + 196 - 3x^2 - 4] / h = (48x + 192) / h
Finally, we divide the numerator by h:
(48x + 192) / h
Therefore, the expression f(x+8) - f(x) / h simplifies to (48x + 192) / h.