For the functions f(x)=x/x+3and g(x)=7/x, find the composition (f∘g)(x) and simplify your answer as much as possible. Write the domain using interval notation
Well, let's form the composition (f∘g)(x) by plugging g(x) into f(x):
(f∘g)(x) = f(g(x))
We know that g(x) = 7/x, so let's replace g(x) in f(x):
f(g(x)) = f(7/x)
Now let's substitute f(x) = x/(x+3) into this expression:
f(7/x) = (7/x) / (7/x + 3)
To simplify further, let's find a common denominator and combine the fractions:
f(7/x) = (7/x) / (7/x + 3) = (7/x) / (7/x + 3x/x) = (7/x) / ((7+3x)/x) = 7 / (7 + 3x)
So, (f∘g)(x) = 7 / (7 + 3x)
Now let's determine the domain using interval notation. Since we have a fraction, we need to make sure the denominator doesn't equal zero:
7 + 3x ≠ 0
Solving this inequality, we find:
3x ≠ -7
x ≠ -7/3
Therefore, the domain of (f∘g)(x) is (-∞, -7/3) U (-7/3, ∞) in interval notation.
Remember, math can be funny, but the honesty is always the best policy!
To find the composition (f∘g)(x), we need to substitute g(x) into f(x).
First, let's find g(x):
g(x) = 7/x
Now, substitute g(x) into f(x):
f(g(x)) = f(7/x)
Next, substitute g(x) = 7/x into f(x) = x/(x + 3):
f(g(x)) = (7/x) / ((7/x) + 3)
To simplify this expression, we need to get rid of the denominator in the denominator. We can do this by multiplying the numerator and denominator by x:
f(g(x)) = (7/x) * (x/(7 + 3x))
Simplifying further:
f(g(x)) = 7 / (7 + 3x)
So, the composition of f(x) and g(x) is f(g(x)) = 7 / (7 + 3x).
Now, let's determine the domain of the composition:
The composition is defined for all x values except those that make the denominator zero. In this case, the denominator is 7 + 3x.
To find the values of x for which the denominator is zero, we solve the equation:
7 + 3x = 0
Subtracting 7 from both sides:
3x = -7
Dividing both sides by 3:
x = -7/3
Therefore, the composition (f∘g)(x) is defined for all x except x = -7/3.
In interval notation, the domain is (-∞, -7/3) U (-7/3, +∞).
To find the composition (f∘g)(x), which represents the composition of functions f and g, we need to substitute the expression for g(x) into f(x).
First, let's express the function f(x) as f(x) = x/(x + 3) and g(x) as g(x) = 7/x.
Now, substitute g(x) into f(x):
(f∘g)(x) = f(g(x)) = f(7/x)
Replace x in f(x) with 7/x:
f(g(x)) = (7/x) / ((7/x) + 3)
To simplify this expression, we can multiply the numerator and denominator by x to eliminate the fraction within a fraction:
f(g(x)) = (7/x) * (x / ((7/x) + 3))
Simplify further:
f(g(x)) = (7x) / (7 + 3x)
The composition (f∘g)(x) is (7x) / (7 + 3x).
To determine the domain of this composition, we need to consider any restrictions on x. In this case, the only potential restriction is due to the presence of x in the denominator.
We know that the denominator, 7 + 3x, cannot be equal to zero since division by zero is undefined. So, we set it non-zero and solve for x:
7 + 3x ≠ 0
3x ≠ -7
x ≠ -7/3
Therefore, the domain of (f∘g)(x) is all real numbers except -7/3, which is expressed in interval notation as (-∞, -7/3) ∪ (-7/3, ∞).
f(g) = g/(g+3) = (7/x) / (7/x + 3) = 7/(3x+7)
exclude from the domain values where the denominator is zero.
also,
since f(x) is undefined for x = -3, exclude values where g(x) = -3
since g(x) is undefined for x=0, exclude that as well