Find the values for the following functions.
If f(x)=2^x, show that f(x+3) -f (x-1)=15/2f(x)
If G(x)= sin 2x, find G(0),G(1/4 pie),and G(7/8 pie)
f(x+3) = 2^(x+3) = 2^x * 2^3 = 8*2^x
f(x-1) = 1/2 * 2^x
PI, not PIE!
G(0) = sin(2*0) = sin(0) = 0
similarly, just plug in the other values for x.
These are some of your standard angles, whose trig functions you should know. They come up so often, it's better if you just take a few minutes and memorize their function values. It will save you lots of time latere, wondering what they are.
To find the values for the given functions, let's go step by step.
For the function f(x) = 2^x:
Step 1: Substitute x+3 and x-1 into the function f(x) to find f(x+3) and f(x-1):
f(x+3) = 2^(x+3)
f(x-1) = 2^(x-1)
Step 2: Subtract f(x-1) from f(x+3):
f(x+3) - f(x-1) = 2^(x+3) - 2^(x-1)
Step 3: Rewrite the right side using exponential properties:
= 2^x * 2^3 - 2^x * 2^(-1)
= 8 * 2^x - (1/2) * 2^x
= (8 - 1/2) * 2^x
= (15/2) * 2^x
Therefore, we have shown that f(x+3) - f(x-1) = (15/2) * f(x)
For the function G(x) = sin(2x):
Step 1: Substitute the given values into the function G(x) to find G(0), G(1/4 π), and G(7/8 π):
G(0) = sin(2 * 0) = sin(0) = 0
G(1/4 π) = sin(2 * (1/4 π)) = sin(1/2 π) = 1
G(7/8 π) = sin(2 * (7/8 π)) = sin(7/4 π) = -1
Therefore, G(0) = 0, G(1/4 π) = 1, and G(7/8 π) = -1.
To find the values for the given functions, let's go through each one step by step.
1. Function f(x) = 2^x:
To show that f(x+3) - f(x-1) = (15/2) * f(x), we need to substitute the given values into the equation and verify if it holds true.
f(x+3) = 2^(x+3) ---> (1)
f(x-1) = 2^(x-1) ---> (2)
f(x) = 2^x ---> (3)
Substituting equations (1), (2), and (3) into f(x+3) - f(x-1), we get:
2^(x+3) - 2^(x-1) = (15/2) * 2^x
To simplify further:
2^x * 2^3 - 2^x * 2^(-1) = (15/2) * 2^x
By applying the rules of exponents:
8 * 2^x - (1/2) * 2^x = (15/2) * 2^x
Combining the like terms on the left side:
(16/2 - 1/2) * 2^x = (15/2) * 2^x
Simplifying the expression:
(15/2) * 2^x = (15/2) * 2^x
Therefore, we have shown that f(x+3) - f(x-1) = (15/2) * f(x) for the function f(x) = 2^x.
2. Function G(x) = sin(2x):
To find the values of G(0), G(1/4 π), and G(7/8 π), we need to substitute the given values of x into the function G(x) = sin(2x) and evaluate.
G(0) = sin(2(0)) = sin(0) = 0
G(1/4 π) = sin(2(1/4 π)) = sin(π/2) = 1
G(7/8 π) = sin(2(7/8 π)) = sin(7π/4) = -1/sqrt(2) or approximately -0.7071