SOLUTION: f(x) = x^2 + 5 a) f(-6) = b) Find if f(x) = 14 c) f(x + h) = d) Calculate: f(x + h) - f(x) over h Can you please explain to me this homework question by showing th

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Question 1203773: f(x) = x^2 + 5
a) f(-6) =
b) Find if f(x) = 14
c) f(x + h) =
d) Calculate: f(x + h) - f(x) over h

Can you please explain to me this homework question by showing the step-by-step calculations? Thank you!
Found 2 solutions by Theo, mananth:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
f(x) = x^2 + 5

a.
f(-6) = (-6)^2 + 5 = 36 + 5 = 41.
x is replaced by -6.

b.
f(x) = 14.
this means that x^2 + 5 = 14
subtract 5 from both sides of the equation to get x^2 = 9.
solve for x to get x = 3
f(3) = 3^2 + 5 = 14.

c.
f(x) = x^2 + 5
f(x+h) = (x+h)^2 + 5
you are replacing (x) with (x+h)
f(x+h) = (x+h)^2 + 5 = x^2 + 2hx + h^2 + 5

d.
Calculate: f(x + h) - f(x) over h
from (c), you have f(x+h) = x^2 + 2hx + h^2 + 5
subtract f(x) from that and divide by h to get:
(f(x+h) - f(x)) / h = (x^2 + 2hx + 5 + h^2 - x^2 - 5) / h
simplify to get (2hx + h^2) / h
simplify further to get 2x + h.

this might be easier to see in the attached worksheet.

the limit as h approaches 0 becomes 2x.
this is the derivative of x^2 + 5.
you will learn this later, if you haven't already.

Answer by mananth(16946) (Show Source):
You can put this solution on YOUR website!

a) find f(-6), plug -6 into the function
f(-6) = (-6)^2 + 5
f(-6) = 36 + 5
f(-6) = 41
b) f(x) = 14, find x:
14 = x^2 + 5
Subtract 5 from both sides:
x^2 = 14 - 5
x^2 = 9
take the square root of both sides:
x = ±sqrt(9)
x = ±3
So, x = 3 and x = -3.
c) To find f(x + h), you plug x + h for x:
f(x + h) = (x + h)^2 + 5
d) We know f(x+h) and f(x) from above
=
simplify the R H S

=

cancel x^2 and -x^2 we get

h(2x + h)/h

So,