Combined/Composite Function Continuity and the

Intermediate Value Theorem

Lesson 1.1.15

Learning Objectives

• Given a combined or composite function and a point, determine if the function is continuous at that point.

• Apply the Intermediate Value Theorem to determine which y-values a continuous function must contain in a certain x-value interval [a, b].

Review of Composite Functions

• A composite function is f(g(x)), or (f○g)(x). We learned this in Lesson 1.1.4.

• Example: If f(x) = x2 + 1, and g(x) = x – 2, determine f(g(3)).

• Now determine f(g(x)).

Continuity of Composite Functions

We say that the composite function f(g(x)) is continuous at a point c if:

• g is continuous at c.

• f is continuous at g(c).

Example 1

Determine the continuity of v(p(s)) where v(s) = s3 and p(s) = 6s2 where s = 1.

• First, we test to see if the function p is continuous at 1 using the three continuity tests from Lesson 1.1.14.

• If it is, take whatever you get for p(1) and see if the function v is continuous at that value. Again, you would use the three tests. Your input for v is p(1), not 1.

Continuity of Combined Functions

Let h(x) be a combination of the functions f(x) and g(x) in some way. If f and g are continuous at c, we say that h is continuous at c as well if the following conditions hold:

• h(x) = f(x) + g(x)• h(x) = f(x) – g(x)• h(x) = f(x) ● g(x)• h(x) = f(x) / g(x) if g(x) ≠ 0.

Example 2

• On which intervals is the following function continuous?

x

x 2

Intermediate Value Theorem

• Challenge: draw a path from a to b without picking up your pencil. You cannot cross the f(c) line. You must stay within the boundaries of a and b. Can you do it?

• Answer: No you can’t, by the Intermediate Value Theorem.

• Relate this idea to continuous functions. If a function is continuous from x coordinates a to b, it must go from y coordinates f(a) to f(b), crossing every y value in between.

More Formally…

• The Intermediate Value Theorem tells us that, if a function f  is continuous on some closed interval [a, b] and k is some value between f (a) and f (b), then there exists an h such that f (h) = k.

Example 3

• Let f be a continuous function on the interval [-3, 5]. If f(-3) = 1, and f(5) = 3, name three y-values that must be on the function in this interval.

Example 4

Prove that the function below has a root in the interval [1, 2].

(A root is an x value that makes f(x) = 0.)

Wrap-Up

• Know how to determine if a composite function and combined function is continuous at a point.

• Know how to apply the Intermediate Value Theorem

Homework

• Reteaching problems.