4. REVIEW: Let f(x) = 2^x and g(x) = √(x − 7). Find and simplify completely.
a. )( f(3+h)-f(3) )/8h
b.) (g ◦ f)(5) / (f ◦ f)(3)
f(3+h) = 2^(3+h) = 8*2^h
f(3) = 2^3 = 8
...
(g◦f)(5) / (f◦f)(3)
= √(f(5)-7) / 2^f(3)
= √(32-7) / 2^8
= 5/64
Who the hell is on here at 1:30 in the morning?
To find and simplify the expressions, let's first determine the values of f(x) and g(x):
Given:
f(x) = 2^x
g(x) = √(x - 7)
a. ( f(3+h) - f(3) ) / 8h:
To find f(3+h), substitute x = 3+h into the function f(x):
f(3+h) = 2^(3+h)
Next, find f(3) by substituting x = 3 into the function f(x):
f(3) = 2^3
Now substitute these values into the given expression:
( f(3+h) - f(3) ) / 8h = (2^(3+h) - 2^3) / 8h
The expression is now simplified, but if you want to simplify it further, you can combine the two terms with exponents using the properties of exponents. The general rule is:
a^m - a^n = a^n * (a^(m-n) - 1)
Applying this rule to our expression, we have:
= (2^3 * 2^h - 2^3) / 8h
= (8 * 2^h - 8) / 8h
= 8 * (2^h - 1) / 8h
= (2^h - 1) / h
b. (g ◦ f)(5) / (f ◦ f)(3):
To find (g ◦ f)(5), we need to evaluate the composition of g(f(5)):
(g ◦ f)(5) = g(f(5)) = g(2^5)
Substituting this value into the function g(x):
g(2^5) = √(2^5 - 7)
= √(32 - 7)
= √25
= 5
To find (f ◦ f)(3), we need to evaluate the composition of f(f(3)):
(f ◦ f)(3) = f(f(3)) = f(2^3)
Substituting this value into the function f(x):
f(2^3) = 2^(2^3)
= 2^8
= 256
Now substitute these values into the given expression:
(g ◦ f)(5) / (f ◦ f)(3) = 5 / 256
The expression is already simplified.
Therefore, the simplified forms are:
a. ( f(3+h) - f(3) ) / 8h = (2^h - 1) / h
b. (g ◦ f)(5) / (f ◦ f)(3) = 5/256