If f(x)= 3/x-4 and g(x)=2x, there is at least one value of x for which the composite function f(g(x)) cannot be evaluated.
What is that value of x? For what value(s) of x is it impossible to evaluate g(f(x))?
How do you figure this out?
As a first step, graph f(x) and find the discontinuity for the function f(x).
Call this point x0.
If this point x0 is found, x0=g(x) is not admissible in the composition f(g(x)), because x0 is not in the domain of f(x).
Since x0=g(x)=2x, we conclude that x=x0/2 is not an admissible value of x in f(g(x)) because
f(g(x0/2)) = f(2(x0/2))=f(x0) which is undefined.
You can find the value of x in the second part using a similar argument.
If you need further help, please post.
To determine the value(s) of x for which the composite function f(g(x)) cannot be evaluated, we need to consider the domain of both f(x) and g(x).
For f(x) = 3/(x - 4), the function is not defined when the denominator (x - 4) becomes zero, as division by zero is undefined. Therefore, the value x = 4 is not in the domain of f(x).
Next, for g(x) = 2x, there are no restrictions on the domain since it is defined for all real numbers.
Now, to evaluate the composite function f(g(x)), we substitute g(x) into f(x). Therefore, we have:
f(g(x)) = f(2x) = 3/(2x - 4)
To find the value of x for which f(g(x)) cannot be evaluated, we need to consider the domain of f(g(x)). Similar to f(x), the denominator (2x - 4) cannot be zero. We solve for x to find when the denominator becomes zero:
2x - 4 = 0
2x = 4
x = 2
Therefore, the value x = 2 is not in the domain of f(g(x)), and thus, f(g(x)) cannot be evaluated at x = 2.
Now, let's consider the composite function g(f(x)). We substitute f(x) into g(x) to find:
g(f(x)) = g(3/(x - 4)) = 2 * (3/(x - 4)) = 6/(x - 4)
To determine the value(s) of x for which g(f(x)) cannot be evaluated, we need to consider the domain of g(f(x)). Again, we observe that the denominator (x - 4) cannot be zero. Solving for x:
x - 4 = 0
x = 4
Therefore, the value x = 4 is not in the domain of g(f(x)), and thus, g(f(x)) cannot be evaluated at x = 4.
In summary:
- The value of x for which f(g(x)) cannot be evaluated is x = 2.
- The value of x for which g(f(x)) cannot be evaluated is x = 4.
To figure this out, we analyzed the domain of both f(x) and g(x), identifying the values that make the denominators zero in each case, as division by zero is undefined. This information allowed us to determine the values that make the composite functions f(g(x)) and g(f(x)) impossible to evaluate.