Find each of the following functions.
f(x) = x2 − 3, g(x) = 5x + 2
(a) f compose g
State the domain of the function. (Enter your answer using interval notation.)
(b) g compose f
State the domain of the function. (Enter your answer using interval notation.)
(c) f compose f
State the domain of the function. (Enter your answer using interval notation.)
(d) g compose g
State the domain of the function. (Enter your answer using interval notation.)
f◦g = f(g) = g^2-3 = (5x+2)^2-3
g◦f = g(f) = 5f+2 = 5(x^2-3)+2
f◦f = f(f) = f^2-3 = (x^2-3)^2-3
g◦g = g(g) = 5g+2 = 5(5x+2)+2
the domain of all polynomials is all real numbers.
To find the composition of two functions, we substitute the second function into the first function.
(a) Find f compose g:
f compose g(x) = f(g(x))
Substituting g(x) into f(x), we get:
f(g(x)) = (g(x))^2 - 3
Since g(x) = 5x + 2, we can rewrite the composition as:
f compose g(x) = (5x + 2)^2 - 3
(b) Find g compose f:
g compose f(x) = g(f(x))
Substituting f(x) into g(x), we get:
g(f(x)) = 5(f(x)) + 2
Since f(x) = x^2 - 3, we can rewrite the composition as:
g compose f(x) = 5((x^2 - 3)) + 2
(c) Find f compose f:
f compose f(x) = f(f(x))
Substituting f(x) into f(x), we get:
f(f(x)) = (f(x))^2 - 3
Since f(x) = x^2 - 3, we can rewrite the composition as:
f compose f(x) = ((x^2 - 3))^2 - 3
(d) Find g compose g:
g compose g(x) = g(g(x))
Substituting g(x) into g(x), we get:
g(g(x)) = 5(g(x)) + 2
Since g(x) = 5x + 2, we can rewrite the composition as:
g compose g(x) = 5((5x + 2)) + 2
Now let's determine the domain of each function:
(a) The domain of f compose g is the set of all x-values for which the expression (5x + 2) is defined and the resulting output is defined in f(x) = x^2 - 3. Since there are no restrictions on the domain for both functions, the domain of f compose g is the set of all real numbers, represented by (-∞, ∞).
(b) The domain of g compose f is the set of all x-values for which the expression (x^2 - 3) is defined and the resulting output is defined in g(x) = 5x + 2. Since there are no restrictions on the domain for both functions, the domain of g compose f is also the set of all real numbers, represented by (-∞, ∞).
(c) The domain of f compose f is the set of all x-values for which the expression (x^2 - 3) is defined and the resulting output is defined in f(x) = x^2 - 3. Since there are no restrictions on the domain, the domain of f compose f is also the set of all real numbers, represented by (-∞, ∞).
(d) The domain of g compose g is the set of all x-values for which the expression (5x + 2) is defined and the resulting output is defined in g(x) = 5x + 2. Again, there are no restrictions on the domain, so the domain of g compose g is the set of all real numbers, represented by (-∞, ∞).