Let
f(x) = 1/x − 2
and
g(x) = 1/x − 3
. Determine the domain of the composite function. (Enter your answer using interval notation.)
g compose f
_____?
& Find the composite function.
(g compose f)(x) = _____?
g(f) = 1/(f-3) = 1/(1/(x-2)-3) = (2-x)/(3x-7)
So, the domain is all reals except where
3x-7 = 0
or
x-2=0 (because f(2) is not defined)
or
f(x) = 3 (because g(3) is not defined)
Oh, the domain of the composite function sounds like a real adventure! Let's solve the mystery together, shall we?
First, let's find the composite function, (g compose f)(x), before we jump into the domain fun.
To find (g compose f)(x), we need to substitute f(x) into g(x). So, let's dive in:
(g compose f)(x) = g(f(x))
Plugging in the function f(x) into g(x), we get:
(g compose f)(x) = 1/(1/x - 2) - 3
Now, let's simplify that expression:
(g compose f)(x) = 1/((1 - 2x) / x) - 3
To avoid any mathematical clownery, let's simplify it further by multiplying the numerator and denominator by x:
(g compose f)(x) = x / (1 - 2x) - 3
Now that we've found the composite function, let's move on to the domain.
The domain refers to the values of x that we can put into the composite function without causing any mathematic mind-bending situations. In this case, we need to avoid any divisory disasters or square root capers.
In the composite function (g compose f)(x), we have a denominator of (1 - 2x). To prevent any division mishaps, we need to make sure the denominator is never equal to zero.
So, to determine the domain, let's solve the inequality:
1 - 2x ≠ 0
Solving that simple equation, we find:
-2x ≠ -1
x ≠ 1/2
So the domain of the composite function is all real numbers except x = 1/2.
In interval notation, we can represent that as:
Domain of (g compose f) = (-∞, 1/2) U (1/2, ∞)
And there you have it! We've determined the domain of the composite function (g compose f) and found the composite function itself. Happy math-ing!
To determine the domain of the composite function g compose f, we need to consider the restrictions on both f(x) and g(x).
The function f(x) can be undefined when the denominator, x, is equal to 0. Therefore, x cannot be equal to 0 for f(x) to be defined.
Now, let's find the composite function (g compose f)(x). This means we need to substitute f(x) into g(x) as the input.
(g compose f)(x) = g(f(x)) = g(1/x - 2)
To simplify the composite function, let's substitute f(x) into g(x).
(g compose f)(x) = g(1/x - 2) = 1/(1/x - 2) - 3
To simplify this expression further, we need to find a common denominator.
(g compose f)(x) = 1/[(1 - 2x)/x] - 3
Next, we can simplify the expression by multiplying the numerator and denominator by x.
(g compose f)(x) = 1/[x(1 - 2x)] - 3
Now, let's determine the domain of the composite function.
Since division by 0 is undefined, we need to find the values of x for which the denominator is equal to 0. In this case, the denominator is x(1 - 2x).
Setting the denominator to 0, we have:
x(1 - 2x) = 0
This equation is satisfied when either x = 0 or 1 - 2x = 0.
Solving 1 - 2x = 0, we get x = 1/2.
Therefore, the values of x for which the denominator is equal to 0 are x = 0 and x = 1/2.
However, x = 0 is already excluded from the domain of f(x). So, we just need to consider x = 1/2.
Thus, the domain of the composite function (g compose f) is all real numbers except x = 0 and x = 1/2.
In interval notation, the domain of (g compose f) is (-∞, 0) ∪ (0, 1/2) ∪ (1/2, ∞).
To determine the domain of the composite function g composed with f, denoted as (g compose f)(x), we need to find the values of x for which the composition is defined.
To do this, we first need to find the composition of the two functions:
(g compose f)(x) = g(f(x))
Substituting f(x) into the function g(x):
g(f(x)) = g(1/x - 2)
Now, let's find the domain of the composite function. The domain of a composite function is determined by the domains of each individual function involved.
The function f(x) = 1/x - 2 is defined for all real numbers except for x = 0, as division by zero is undefined. So, the domain of f is (-∞, 0) U (0, ∞).
The function g(x) = 1/x - 3 is also defined for all real numbers except for x = 0, as division by zero is again undefined. So, the domain of g is (-∞, 0) U (0, ∞).
Now, we need to determine the values of x for which both f(x) and g(f(x)) are defined.
Since the domain of f is (-∞, 0) U (0, ∞), and the domain of g is also (-∞, 0) U (0, ∞), the only value to exclude from the composite function's domain is x = 0.
Therefore, the domain of the composite function (g compose f)(x) is (-∞, 0) U (0, ∞), excluding x = 0. This can be written in interval notation as (-∞, 0) U (0, ∞).
So, the domain of (g compose f)(x) is (-∞, 0) U (0, ∞).
Now, to find the explicit expression for (g compose f)(x), we substitute f(x) into g(x):
(g compose f)(x) = g(f(x))
= g(1/x - 2)
= 1/(1/x - 2) - 3
Simplifying this expression, we get:
= 1/((1 - 2x)/x) - 3
= x/(1 - 2x) - 3
So, the explicit expression for (g compose f)(x) is x/(1 - 2x) - 3.