Evaluate Composition of Function and Limit of Piecewise Function

Math Problem Statement

A function f has the property that f(0) = 1, f(1) = -1, f(-1) = 2, and f(2) = 0. Evaluate the composition f(f(f(0))). Let f(x) be defined by f(x) = x, if x >= 0, and f(x) = 1, if x < 0. Find the limit of f(x) as x approaches 0.

Solution

Let's break down the problems from the image you've uploaded.

a) The function $f$ has the following properties:

$f(0) = 1$
$f(1) = -1$
$f(-1) = 2$
$f(2) = 0$

We are asked to evaluate the composition $f(f(f(0)))$ .

Solution:

Start by finding $f(0)$ , which is given: $f(0) = 1$
Now find $f(f(0)) = f(1)$ . From the problem, we know: $f(1) = -1$
Finally, find $f(f(f(0))) = f(-1)$ . We are given: $f(-1) = 2$ Thus, the value of $f(f(f(0)))$ is $2$ .

b) The function $f(x)$ is defined as:

x, & \text{if } x \geq 0 \\ 1, & \text{if } x < 0 \end{cases}$$ We are asked to find the limit: $$\lim_{x \to 0} f(x)$$ #### Solution: To evaluate this limit, we must examine the left-hand limit (as $$x \to 0^-$$) and the right-hand limit (as $$x \to 0^+$$) separately. 1. **Right-hand limit** $$\lim_{x \to 0^+} f(x)$$: - For $$x \to 0^+$$ (values of $$x$$ approaching 0 from the positive side), $$f(x) = x$$. So: $$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0$$ 2. **Left-hand limit** $$\lim_{x \to 0^-} f(x)$$: - For $$x \to 0^-$$ (values of $$x$$ approaching 0 from the negative side), $$f(x) = 1$$. So: $$\lim_{x \to 0^-} f(x) = 1$$ Since the left-hand and right-hand limits are not equal, the limit $$\lim_{x \to 0} f(x)$$ does **not exist**. ### Summary: - For part (a), $$f(f(f(0))) = 2$$. - For part (b), the limit $$\lim_{x \to 0} f(x)$$ does not exist because the left-hand and right-hand limits are different. --- Do you need further details or explanations on any part? Here are some related questions to expand your understanding: 1. How do you evaluate composite functions step by step? 2. What conditions are necessary for a limit to exist at a point? 3. What does it mean for a function to be continuous at a point? 4. How can you graphically interpret the difference between left-hand and right-hand limits? 5. Can a function be discontinuous at a point if both the left-hand and right-hand limits exist but are different? **Tip:** When finding limits, always check both the left-hand and right-hand limits separately for functions that have different behaviors on either side of a point.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Limits
Piecewise Functions

Formulas

Theorems

Limit Theorem
Definition of Function Composition

Suitable Grade Level

Grades 11-12

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