Find a. fog b. gof c. fof d. gog
f(x)=4-x
g(x)=3x-5
To find the compositions a. fog, b. gof, c. fof, and d. gog, we first need to understand the concept of function composition.
Function composition is the process of combining two functions to create a new function. It is denoted by writing one function followed by another function, with brackets indicating the order of operations.
a. fog: To find a. fog, we need to substitute the expression for g(x) into f(x).
fog(x) = f(g(x)) = f(3x-5)
To get the result, we substitute 3x-5 into f(x):
fog(x) = 4 - (3x-5) = 4 - 3x + 5 = 9 - 3x
Therefore, a. fog = 9 - 3x.
b. gof: To find b. gof, we substitute the expression for f(x) into g(x).
gof(x) = g(f(x)) = g(4-x)
To get the result, we substitute 4-x into g(x):
gof(x) = 3(4-x) - 5 = 12 - 3x - 5 = 7 - 3x
Therefore, b. gof = 7 - 3x.
c. fof: To find c. fof, we substitute the expression for f(x) into itself.
fof(x) = f(f(x)) = f(4-x)
To get the result, we substitute 4-x into f(x):
fof(x) = 4 - (4-x) = 4 - 4 + x = x
Therefore, c. fof = x.
d. gog: To find d. gog, we substitute the expression for g(x) into itself.
gog(x) = g(g(x)) = g(3x-5)
To get the result, we substitute 3x-5 into g(x):
gog(x) = 3(3x-5) - 5 = 9x - 15 - 5 = 9x - 20
Therefore, d. gog = 9x - 20.
So, the compositions are:
a. fog = 9 - 3x
b. gof = 7 - 3x
c. fof = x
d. gog = 9x - 20.