Let f(x) = 2x and g(x) = √(x − 7). Find and simplify completely
a.) f(3+h)−f(3)/ 8h
b.) g ◦ f)(5)/ (f ◦ f)(3)
for part 8 i have,
2^(3+h) - 2^3/8h = 8*2^h -8 / 8h.
I'm having a brain fart but am i able to subtract 8 from 8 or is the 8*2^h seen as one value, and how can i get rid of the 8h on the bottom.
for part B, i have
√(2^5 − 7) /2^(2^3) = √(32 − 7)/ 2^8 = √25/256 = 5/256.
I posted this question earlier and as the bottom they had it as 64. Im wondering if they were correct with the 64 or am i with the 256?
for part a i messed up. my final result i got is 1/8(2^(3+h) -8)h
(2^(3+h) - 2^3)/(8h)
= (8*2^h -8)/(8h)
= 8(2^h - 1)/(8h)
= (2^h - 1)/h
f(x) = 2^x
g(x) = √(x-7)
(g◦f)(5)/(f◦f)(3)
= √(f(5)-7)/2^(f(3))
= √(2^5-7)/2^(2^3)
= √(32-7)/2^8
= 5/256
Let's go through each part step by step:
a.) To simplify the expression f(3+h) - f(3) / 8h, we will substitute the values of f(x) into the expression:
f(3+h) - f(3) = 2(3+h) - 2(3) = 6 + 2h - 6 = 2h
So the expression becomes 2h / 8h. Now, to simplify further, we can cancel out the common factor of h:
2h / 8h = 2 / 8 = 1 / 4
Therefore, the simplified expression is 1 / 4.
b.) To simplify the expression g ◦ f(5) / (f ◦ f)(3), we will substitute the values of g(x) and f(x) into the expression:
g ◦ f(5) means we first evaluate f(5), which is 2 * 5 = 10, and then substitute this value into g(x):
g ◦ f(5) = g(10) = √(10 - 7) = √3
Next, we need to calculate (f ◦ f)(3), which means we first evaluate f(3), which is 2 * 3 = 6, and then substitute this value into f(x):
(f ◦ f)(3) = f(f(3)) = f(6) = 2 * 6 = 12
So the expression becomes √3 / 12. To simplify further, we can find the prime factorization of 12: 12 = 2^2 * 3. Therefore:
√3 / 12 = √3 / (2^2 * 3) = √3 / (4 * 3) = √3 / 12
From your calculations, it seems that you have simplification errors in both parts:
a.) The correct answer is 1 / 4, not 8 * 2^h - 8 / 8h.
b.) The correct answer is √3 / 12, not 5 / 256 or 5 / 64.
So, it seems that the previous answer with 64 is incorrect, and you have obtained the correct answer with 256.