Let f(x)=SQRT X+1 and g(x)=3x−5. Find f∘g and g∘f, and their respective domains.

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):

you know that √u has domain u>=0

f(g) = √(g+1) = √(3x-4)
so, its domain is 3x-4 >= 0

g(f) = 3f-5 = 3√(x+1)-5
same domain as f, naturally: x+1 >= 0

To find f∘g, we need to substitute g(x) into f(x).

1. f∘g(x) = f(g(x))
= f(3x−5)

To find g∘f, we need to substitute f(x) into g(x).

2. g∘f(x) = g(f(x))
= g(√(x+1))

Now, let's find the respective domains for f∘g and g∘f.

To determine the domain of f∘g, we need to consider the domain of g(x) = 3x−5. Since g(x) is a linear function, it is defined for all real numbers.

Therefore, the domain of f∘g is also all real numbers.

3. (g∘f)(x) = g(f(x))
= g(√(x+1))
= 3(√(x+1))−5

To determine the domain of g∘f, we need to consider the domain of f(x) = √(x+1).

Since the square root function is defined only for non-negative real numbers, the domain of f(x) is x ≥ -1.

Therefore, the domain of g∘f is all real numbers greater than or equal to -1.

Answer:

1. Domain of f∘g: (-∞, ∞)

3. (g∘f)(x) = 3(√(x+1))−5

4. Domain of g∘f: [-1, ∞)

To find the composition f∘g, we need to substitute the function g into f.

1. f∘g(x) = f(g(x)) = f(3x - 5)
Let's substitute g(x) = 3x - 5 into f(x) = √(x + 1):
f(3x - 5) = √((3x - 5) + 1) = √(3x - 4)
Therefore, f∘g(x) = √(3x - 4).

Now let's find the domain of f∘g:

2. To determine the domain of f∘g, we need to consider the domain restrictions of both functions f(x) and g(x).

- The domain of f(x) = √(x + 1) is restricted by the square root operation. So, the expression inside the square root must be greater than or equal to zero:
x + 1 >= 0
x >= -1

- The domain of g(x) = 3x - 5 is unrestricted since it is a linear function.

3. Therefore, the domain of f∘g is given by the domain of g(x), which is all real numbers, since there are no additional domain restrictions applied when substituting g(x) into f(x).
Domain of f∘g: (-∞, ∞) or (-∞, +∞) in interval notation.

Now let's find g∘f:

4. g∘f(x) = g(f(x)) = g(√(x + 1))
Let's substitute f(x) = √(x + 1) into g(x) = 3x - 5:
g(√(x + 1)) = 3(√(x + 1)) - 5 = 3√(x + 1) - 5
Therefore, g∘f(x) = 3√(x + 1) - 5.

To determine the domain of g∘f, we need to consider the domain restrictions again:

5. The domain of f(x) = √(x + 1) is the same as we found earlier: x >= -1.

6. The domain of g(x) = 3x - 5 is still unrestricted since it is a linear function.

7. Therefore, the domain of g∘f is given by the domain of f(x), which is x >= -1, since there are no additional domain restrictions applied when substituting f(x) into g(x).
Domain of g∘f: [-1, ∞) in interval notation.