Question 1203773
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.


<img src = "http://theo.x10hosting.com/2023/092101.jpg">


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.