Question

A piecewise function is shown below

Answer 1

Answer:

p = 5/7

Step-by-step explanation:

The given function is:

$g(x) = -3x^{2} - 2x + 8$ for -4 ≦ x < 1

$g(x) = -2x + 7p$ for 1 ≦ x ≦ 5

Part a)

A continuous function has no breaks, jumps or holes in it. So, in order for g(x) to be continuous, the point where g(x) stops during the first interval -4 ≦ x < 1 must be equal to the point where g(x) starts in the second interval 1 ≦ x ≦ 5

The point where, g(x) stops during the first interval is at x = 1, which will be:

$-3(1)^{2}-2(1)+8=3$

The point where g(x) starts during the second interval is:

$-2(1)+7(p) = 7p - 2$

For the function to be continuous, these two points must be equal. Setting them equal, we get:

3 = 7p - 2

3 + 2 = 7p

p = $\frac{5}{7}$

Thus the value of p for which g(x) will be continuous is $\frac{5}{7}$.

Part b)

We have to find p by setting the two pieces equal to each other. So, we get the equation as:

$-3x^{2}-2x+8=-2x+7p\\\\ -3x^{2}+8=7p$

Substituting the point identified in part (a) i.e. x=1, we get:

$-3(1)^{2}+8=7p\\\\ 5=7p\\\\ p=\frac{5}{7}$

This value agrees with the answer found in previous part.