https://file.aiquickdraw.com/mathbot/file/1727226930902q99qha4a.jpg

The function \( f(x) \) is defined as follows:

- \( f(x) = 3x - 4 \) for \( 3 \leq x \leq 4 \)
- \( f(x) = x^3 - 4 \) for \( 4 < x \leq 5 \)

Let's calculate \( f(0) \), \( f(1) \), \( f(4) \), and \( f(5) \):

1. **\( f(0) \)**: Since \( 0 \) is less than \( 3 \), and there's no definition provided for \( f(x) \) when \( x < 3 \), \( f(0) \) is undefined.
2. **\( f(1) \)**: Similarly, \( 1 \) is also less than \( 3 \), so \( f(1) \) is also undefined.
3. **\( f(4) \)**: For \( x = 4 \), we use the function \( f(x) = 3x - 4 \) because \( x \) exactly equals \( 4 \) and this interval is inclusive. Thus:
   \[
   f(4) = 3(4) - 4 = 12 - 4 = 8
   \]
4. **\( f(5) \)**: For \( x = 5 \), we use the function \( f(x) = x^3 - 4 \) because \( x \) falls within the interval \( 4 < x \leq 5 \). Thus:
   \[
   f(5) = 5^3 - 4 = 125 - 4 = 121
   \]

### Summary:
- \( f(0) \) is undefined
- \( f(1) \) is undefined
- \( f(4) = 8 \)
- \( f(5) = 121 \)

Do you have any other questions or need further details on this problem? Here are some follow-up topics we can explore:

1. What would \( f(x) \) be if \( x = 3 \)?
2. How would \( f(x) \) change if \( x \) were slightly above \( 4 \), like \( 4.01 \)?
3. Can we graph the function \( f(x) \) to visually understand the values across different intervals?
4. How does the discontinuity between the piecewise segments at \( x = 4 \) affect the function graphically?
5. What's the general approach to solving piecewise functions like this one?

**Tip:** When dealing with piecewise functions, always pay close attention to the intervals and their boundaries to determine which expression to use.

If f(x) = {3x - 4 if -3 ≤ x ≤ 4, x^3 - 4 if 4 < x ≤ 5}, find: (a) f(0), (b) f(1), (c) f(4), and (d) f(5).

Learn how to evaluate the piecewise function f(x) defined as 3x - 4 for -3 ≤ x ≤ 4 and x^3 - 4 for 4 < x ≤ 5. This problem explores function evaluation at points x = 0, x = 1, x = 4, and x = 5, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation