f(x) =
1/(1−x) is given. From this we produce two new functions:
g(x) = f(f(x)) and h(x) = f(g(x)).
(a) For which values of x is f(x) undefined? (b) Find a formula for g(x), and simplify it. (c) For which values of x is g(x) undefined? (d) Find a formula for h(x), and simplify it. (e) For which values of x is h(x) undefined?
(Hint: Be careful, because there is a trap here. If a value of x won’t work for f, then it also won’t work for g, because you need to compute f before you compute g!)
g(x) = f(f) = 1/(1-f) = 1/(1-(1/(1-x)) = 1 - 1/x
h(x) = f(g) = 1/(1-g) = 1/(1-(1 - 1/x)) = x
But, as you say, watch for places where f is undefined. I'm sure you can figure those out.
sorry but I don't get the part that I should watch for places where f is undefined. I tried to get the value of x that is f(x) undefined and I got x=1
To determine the values of x for which f(x) is undefined, we need to find the values that make the denominator of f(x) equal to zero. In this case, the denominator is (1 - x). So, we set the denominator equal to zero and solve for x:
1 - x = 0
Adding x to both sides, we get:
1 = x
Therefore, f(x) is undefined for x = 1.
Now, let's find a formula for g(x) by substituting f(x) into itself:
g(x) = f(f(x)) = f(1/(1 - x))
To simplify g(x), we substitute 1/(1 - x) into f(x):
g(x) = f(f(x)) = f(1/(1 - x)) = 1/(1 - (1/(1 - x)))
To simplify the expression, we can combine the fractions:
g(x) = 1/((1 - x)/(1 - x) - 1/(1 - x))
Now, we have a common denominator:
g(x) = 1/((1 - x - 1)/(1 - x)) = 1/( - x/(1 - x)) = -(1 - x)/x
So, a formula for g(x) is g(x) = -(1 - x)/x.
Next, let's determine the values of x for which g(x) is undefined. Since g(x) involves the expression -(1 - x)/x, it will be undefined when the denominator x equals zero. Therefore, g(x) is undefined for x = 0.
Now, let's find a formula for h(x) using the function g(x) we found earlier:
h(x) = f(g(x)) = f(-(1 - x)/x)
Substituting -(1 - x)/x into f(x):
h(x) = f(g(x)) = f(-(1 - x)/x) = 1/(1 - (-(1 - x)/x))
Simplifying the expression:
h(x) = 1/(1 + (1 - x)/x)
To simplify further, let's combine fractions:
h(x) = 1/((x + (1 - x))/x) = 1/(1/x) = x
Thus, a formula for h(x) is h(x) = x.
Now, let's determine the values of x for which h(x) is undefined. Since h(x) is simply equal to x, it is defined for all real values of x. Hence, h(x) is defined for all x.
To summarize:
(a) f(x) is undefined for x = 1.
(b) g(x) = -(1 - x)/x.
(c) g(x) is undefined for x = 0.
(d) h(x) = x.
(e) h(x) is defined for all x.