Let f(x)=x+1‾‾‾‾‾√ and g(x)=3x−5.
Find f∘g and g∘f, and their respective domains.
1. (f∘g)(x)=
2. What is the domain of f∘g ?
Answer (in interval notation):
3. (g∘f)(x)=
4. What is the domain of g∘f ?
Answer (in interval notation):
Help please, don't understand this! thanks
Did you mean f(x) = √(x+1) ??
I will assume that.
f∘g = f(g(x))
= f(3x-5)
= √((3x-5) - 5)
= √(3x - 10)
domain:
3x-10≥0
3x≥10
x ≥ 10/3
( I will let you switch this to "interval notation, I like the "old way" better)
g∘f = g(f(x))
= g(√(x+1) )
=3√(x+1) - 5
domain: x ≥ -1
To find the composition of two functions, f∘g and g∘f, we substitute the expression of one function into the other function. We start by finding (f∘g)(x) and its domain:
1. (f∘g)(x) means to substitute g(x) into f(x). So, we replace x in f(x) with g(x):
(f∘g)(x) = f(g(x)) = f(3x - 5)
Now, let's substitute the expression for g(x) into f(x):
(f∘g)(x) = (3x - 5) + 1√
Simplifying further, we have:
(f∘g)(x) = 3x - 4√
2. To find the domain of f∘g, we need to consider the domains of f(x) and g(x). Given that g(x) = 3x - 5, its domain is all real numbers since there are no restrictions.
Now, we need to find the domain of f(g(x)) = f(3x - 5). The square root function has a domain that includes all real numbers greater than or equal to zero. Therefore, the function f(g(x)) = f(3x - 5) is defined only when 3x - 5 is greater than or equal to zero. Solving this inequality:
3x - 5 ≥ 0
3x ≥ 5
x ≥ 5/3
Hence, the domain of f∘g is [5/3, ∞).
Now, let's find (g∘f)(x) and its domain:
3. (g∘f)(x) means to substitute f(x) into g(x). So, we replace x in g(x) with f(x):
(g∘f)(x) = g(f(x)) = g(x + 1√)
Now, substitute the expression for f(x) into g(x):
(g∘f)(x) = 3(x + 1√) - 5
Simplifying further, we have:
(g∘f)(x) = 3x + 3√ - 5
4. To find the domain of g∘f, we need to consider the domains of f(x) and g(x). With f(x) = x + 1√, there are no restrictions on the domain since taking the square root of a non-negative number is defined for all real numbers.
Hence, the domain of g∘f is (-∞, ∞).