Determine whether fog and gof are continuous at x = 0.

Question

Let $f(x) = x^2$

and

$g(x) = {−4 if x ≤ 0

{|x − 4| if x > 0$

Determine whether fog and gof are continuous at x = 0

Answer 1

f•g(x) = (-4)² if x ≤ 0

= |x - 4|² if x > 0

At 0 the limit from left and right are both 16, and so is the function value f•g(0).

Therefore f•g is continuous.

g•f(x) = -4 if x² ≤ 0

= |x² - 4| if x² > 0

At 0 the limit from left and right are both 16, but the function value g•f(0) = -4

Therefore the function g•f is not continuous.

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It may be less confusing if I rewrite the definition of g using a different variable u

g(u) = -4 if u ≤ 0

|u - 4| if u > 0

Then the calculation of y = g•f(x) can be expressed in two steps

u = f(x)

y = g(u)

For example, suppose we want to calculate y = g•f(0):

u = f(0) = 0² = 0

y = g(0) = -4, because u = 0 ≤ 0

If you want to express g•f using one equation, which is a combination of

the two equations for f and g, then simply replace all occurences of u by x² in the

definition of g(u).

I am sorry, I made a mistake in the limit.

The limit of g•f(x) as x --> 0 is 4, not 16.

Do you still need an explanation of that?