Question

A piecewise function g(x) is defined by g of x is equal to the piecewise function of x cubed minus 16 times x for x is less than 4 and negative log base 4 of the quantity x minus 3 end quantity plus 2 for x is greater than or equal to 4Part A: Graph the piecewise function g(x) and determine the domain. (5 points)Part B: Determine the x-intercepts of g(x). Show all necessary calculations. (5 points)Part C: Describe the interval(s) in which the graph of g(x) is positive. (5 points)

Answer 1

A)

Start by graphing the function on the different intervals.

For this, we graph as if they were independent functions and then join them on the same graph.

for x<4

for x>=4

then, for g(x)

B)

To find the x-intercepts:

equal g(x) to 0.

for x<4

$\begin{gathered} x^3-16x=0 \\ x(x^2-16)=0 \\ \text{equal both factors to 0} \\ 1.x=0 \\ 2.x^2-16=0 \\ \text{then,} \\ x^2-16=0 \\ x^2=16 \\ x=\sqrt[]{16} \\ x=\pm4 \\ 4\text{ is not defined on the interval } \\ x-\text{intercepts are 0 and -4} \end{gathered}$

for x>=4

$\begin{gathered} -\log _4(x-3)+2=0_{} \\ -\log _4(x-3)=-2 \\ \log _4(x-3)=2 \\ 4^{\log _4(x-3)}=4^2 \\ x-3=16 \\ x=19 \\ \text{the x intercept is 19} \end{gathered}$

then, for g(x)

For g(x) the x-intercepts are -4, 0 and 19

C)

finally, using the graph we can see that the function is positive in the intervals [-4,0)∪[4,19) because in these intervals g(x) is above the x-axis.