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Part 1
If f left parenthesis x right parenthesis equals left brace Start 2 By 1 Matrix 1st Row 1st Column 3 x minus 2 if minus 3 less than or equals x less than or equals 5 2nd Row 1st Column x cubed minus 5 if 5 less than x less than or equals 6 EndMatrix
​, ​find: (a)​ f(0), (b)​ f(1), (c) ​f(5​), and​ (d) ​f(6​).

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

\[
f(x) = 
\begin{cases} 
3x - 2 & \text{if } -3 \leq x \leq 5 \\
x^3 - 5 & \text{if } 5 < x \leq 6 
\end{cases}
\]

Now, let's evaluate the function for each specified value of \( x \):

### (a) \( f(0) \):
Since \( 0 \) falls within the interval \( -3 \leq x \leq 5 \), we use the first part of the piecewise function:
\[
f(0) = 3(0) - 2 = -2
\]

### (b) \( f(1) \):
Since \( 1 \) is also within the interval \( -3 \leq x \leq 5 \), we use the same part of the function:
\[
f(1) = 3(1) - 2 = 3 - 2 = 1
\]

### (c) \( f(5) \):
Since \( 5 \) is exactly on the boundary \( x = 5 \), and this value is included in the first interval \( -3 \leq x \leq 5 \), we use the first part of the function:
\[
f(5) = 3(5) - 2 = 15 - 2 = 13
\]

### (d) \( f(6) \):
Since \( 6 \) falls within the interval \( 5 < x \leq 6 \), we use the second part of the piecewise function:
\[
f(6) = 6^3 - 5 = 216 - 5 = 211
\]

### Final Answers:
- (a) \( f(0) = -2 \)
- (b) \( f(1) = 1 \)
- (c) \( f(5) = 13 \)
- (d) \( f(6) = 211 \)

Would you like further explanation on any of the steps?

### Relative Questions:
1. How do you graph piecewise functions like this?
2. What happens if a piecewise function has overlapping intervals?
3. Can a piecewise function be discontinuous?
4. How would you calculate \( f(x) \) for values outside the defined intervals?
5. What is the difference between a piecewise function and a continuous function?

### Tip:
Always double-check which interval a given value falls into when working with piecewise functions.

If f(x) = { 3x - 2 if -3 <= x <= 5, x^3 - 5 if 5 < x <= 6 }, find: (a) f(0), (b) f(1), (c) f(5), and (d) f(6).

Learn how to evaluate a piecewise function where f(x) is defined by two different expressions: a linear function for -3 <= x <= 5 and a cubic function for 5 < x <= 6. This solution explains the step-by-step process of finding f(0), f(1), f(5), and f(6). Suitable for high school students in Grades 9-12.

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