...for what value(s) of x is g(f(x)) undefined? Explain by making reference to the algebraic properties

Question

If f(x) = log x – 4 and g(x) = , for what value(s) of x is g(f(x)) undefined? Explain by making reference to the algebraic properties.

Grade 12 Advanced Functions

Please answer clearly and neatly. Thanks :)

Answer 1

From the given definitions of f(x) and g(x) and the definition of a composite function, we find that

g(f(x)) = g(log(x) - 4)

= 1/((log(x) - 4) + 4)

= 1/log(x).

Now note that there are two different ways that values of x can make g(f(x)) undefined:

1. A value of x can make f(x) undefined.
2. A value of x can result in a value of f(x) that makes g(x) undefined.

From the definition of f(x), we can see that a value of x makes f(x) undefined whenever that value of x makes log(x) undefined. Since logarithm functions are undefined for all values of x <= 0, we also know that f(x) is undefined for all values of x <= 0. We can also see from the preceding definitions of f(x), g(x), and f(g(x)) that a value of x results in a value of f(x) that makes g(x) undefined whenever a value of x makes log(x) = 0. Since log(x) = 0 implies that x = 10^0 = 1, we can conclude that x = 1 results in a value of f(x) that makes g(x) undefined.

Thus, all of the preceding work tells us that the values of x that make g(f(x)) undefined are contained in the solution set {x: x<=0 or x = 1}.