The application of one function followed by the application of a second function to the result of the first as in F^-1(f(x)) is called composition of functions. The two functions need not be inverse of each other. In a diagram, the output of one function is f(x) is used as the input for a second function g(x). In this case, the composite is denoted g(f(x)) which is read "g composed with f of x" or "g of f of x."

**If you could help with anything I'd really appreciate it!! Thank you**

Suppose f and g are functions defined by the following table:
x) -3 -2 -1 0 1 2 3
f(x) 10 8 7 3 -4 -7 -8
g(x) 3 1 0 2 -3 -1 -2

i. Make a table of values for f(g(x)) with x= -3,-2,-1,0,1,2,3 Remember to always start with the innermost set of parentheses

ii. Make a table of values for g(g(x)) with x= -3,-2,-1,0,1,2,3.

iii. Explain why you cannot find the composite function g(f(x)).

a)Consider the two functions f(x)=-x^2+ 7 and g(x)=3x+4. Using input values of -1,2,0.5,and 8 for x:
1. Calculate f(g(x)) for each input value
2. Calculate g(f(x)) for each input value.

B) if f and g are two functions, what must be true about the domains and ranges of f and g in order for f(g(x)) to make sense? What must be true for g(f(x))to make sense?

i. To find the value of f(g(x)), we need to substitute the values of x into g(x) first.

For x = -3:
g(-3) = 3

For x = -2:
g(-2) = 1

For x = -1:
g(-1) = 0

For x = 0:
g(0) = 2

For x = 1:
g(1) = -3

For x = 2:
g(2) = -1

For x = 3:
g(3) = -2

Now, substitute these values into f(x):

For x = 3:
f(g(3)) = f(-2) = -8

For x = 2:
f(g(2)) = f(-1) = -7

For x = 1:
f(g(1)) = f(-3) = -4

For x = 0:
f(g(0)) = f(2) = 7

For x = -1:
f(g(-1)) = f(0) = 3

For x = -2:
f(g(-2)) = f(1) = 8

For x = -3:
f(g(-3)) = f(3) = 10

So, the table of values for f(g(x)) is:
x: -3 -2 -1 0 1 2 3
f(g(x)): 10 8 3 7 -4 -7 -8

ii. To find the value of g(g(x)), we substitute the values of x into g(x) twice:

For x = -3:
g(g(-3)) = g(3) = -2

For x = -2:
g(g(-2)) = g(1) = -3

For x = -1:
g(g(-1)) = g(0) = 2

For x = 0:
g(g(0)) = g(2) = -1

For x = 1:
g(g(1)) = g(-3) = -2

For x = 2:
g(g(2)) = g(-1) = 3

For x = 3:
g(g(3)) = g(-2) = 1

So, the table of values for g(g(x)) is:
x: -3 -2 -1 0 1 2 3
g(g(x)): -2 -3 2 -1 -2 3 1

iii. We cannot find the composite function g(f(x)) because for the composition of functions, the range of the first function must be a subset of the domain of the second function. In this case, the range of f(x) contains values that are not in the domain of g(x) (-4, -7, -8). Therefore, we cannot perform the composition g(f(x)).

a)

1. Calculate f(g(x)):

For x = -1:
g(-1) = 3
f(3) = -3^2 + 7 = -2

For x = 2:
g(2) = -1
f(-1) = -(-1)^2 + 7 = 6

For x = 0.5:
g(0.5) = 0
f(0) = -(0)^2 + 7 = 7

For x = 8:
g(8) = -2
f(-2) = -(-2)^2 + 7 = 3

The values of f(g(x)) for each input value are:
x: -1 2 0.5 8
f(g(x)): -2 6 7 3

2. Calculate g(f(x)):

For x = -1:
f(-1) = -(-1)^2 + 7 = 6
g(6) = -1(6) + 4 = -2

For x = 2:
f(2) = -(2)^2 + 7 = 3
g(3) = -1(3) + 4 = 1

For x = 0.5:
f(0.5) = -(0.5)^2 + 7 = 6.75
g(6.75) = -1(6.75) + 4 ≈ -2.75

For x = 8:
f(8) = -(8)^2 + 7 = -57
g(-57) = -1(-57) + 4 ≈ 61

The values of g(f(x)) for each input value are:
x: -1 2 0.5 8
g(f(x)): -2 1 -2.75 61

b) For f(g(x)) to make sense, the range of g(x) must be a subset of the domain of f(x). In other words, the values produced by g(x) must be valid inputs for f(x).

For g(f(x)) to make sense, the range of f(x) must be a subset of the domain of g(x). In other words, the values produced by f(x) must be valid inputs for g(x).