Math Problem Statement
Question content area top 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).
Solution
The given function is a piecewise function defined as follows:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Cubic Functions
Evaluating Functions
Formulas
f(x) = 3x - 2 for -3 <= x <= 5
f(x) = x^3 - 5 for 5 < x <= 6
Theorems
-
Suitable Grade Level
Grades 9-12