Math

QuestionGraph the piecewise function p(x)p(x) defined as:
p(x)=2x+6+7p(x) = -2|x+6| + 7 for 8x<3-8 \leq x < -3, 13x2\frac{1}{3} x - 2 for 3<x3-3 < x \leq 3, and (x5)25(x-5)^{2} - 5 for 3<x<83 < x < 8.

Studdy Solution

STEP 1

Assumptions1. The function p(x)p(x) is a piece-wise function defined in three parts. . Each part of the function is defined for a specific range of xx values.
3. We need to graph each part of the function within its specified range.

STEP 2

To graph a piece-wise function, we graph each piece separately on the same set of axes. Let's start with the first piece, 2x+6+7-2|x+6|+7 for 8x<-8 \leq x<-.

STEP 3

The absolute value function x|x| is always non-negative. It is defined as xx if x0x\geq0 and x-x if x<0x<0. So, we can rewrite 2x+6+7-2|x+6|+7 as follows2x+6+7={2(x+6)+7,x+602(x6)+7,x+6<0-2|x+6|+7 = \begin{cases} -2(x+6)+7, & x+6\geq0 \\ -2(-x-6)+7, & x+6<0\end{cases}

STEP 4

olve the inequality x+60x+6\geq0 to find the range of xx for which 2(x+6)+7-2(x+6)+7 is defined.
x+60x6x+6\geq0 \Rightarrow x\geq-6So, 2(x+6)+7-2(x+6)+7 is defined for x6x\geq-6.

STEP 5

Similarly, solve the inequality x+<0x+<0 to find the range of xx for which 2(x)+7-2(-x-)+7 is defined.
x+<0x<x+<0 \Rightarrow x<-So, 2(x)+7-2(-x-)+7 is defined for x<x<-.

STEP 6

However, we need to consider the original range for this piece of the function, which is 8x<3-8 \leq x<-3. So, we graph 2(x+6)+-2(x+6)+ for 6x<3-6 \leq x<-3 and 2(x6)+-2(-x-6)+ for 8x<6-8 \leq x<-6.

STEP 7

Now, let's graph the second piece of the function, 13x2\frac{1}{3} x-2 for 3<x3-3<x \leq3. This is a linear function and can be graphed directly within the given range.

STEP 8

Finally, let's graph the third piece of the function, (x5)25(x-5)^{2}-5 for 3<x<83<x<8. This is a quadratic function and can be graphed directly within the given range.

STEP 9

After graphing each piece of the function within its specified range, we combine the graphs to get the graph of the piece-wise function p(x)p(x).

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord