Question

Which of the following piecewise functions is graphed above?

Answer 1

Hello!

The answer is:

The piecewise function that represents the graph, is the option A (first option):

f(x) (piecewise function):

$8; x\leq -1\\\\x^{2} -4x+1;-1<x<5\\\\-x+1\geq 5$

Why?

To find the correct option, we need to look for the piecewise function that contains the following functioncs existing in the determined domains (inputs).

From the graph, we know that we need the following functions:

- A horizontal line, which exists from -∞ to -1, givind as input 8.

The function will be:

$y=8$

Then, the piecewise function it will be:

$8; x\leq -1$

- A quadratic function (convex parabola) which y-intercept is equal to 1, exists from -1 to 5, and it vertex (lowest point for this case) is located at (2,-3)

The function will be:

$y=x^{2}-4x+1$

Finding the y-intercept, we have:

$y=0^{2}-4*80)+1$

$y=1$

Finding the vertex of the parabola, we have:

$x_{vertex}=\frac{-b}{2}\\\\x_{vertex}=\frac{-(-4)}{2}=\frac{4}{2}=2$

$y_{vertex}=x_{vertex}^{2}-4x_{vertex}+1$

$y_{vertex}=2^{2}-4*2+1=4-8+1=-3$

The vertex of the parabola is located at the point (2,-3).

Then, for the piecewise function it will be:

$x^{2} -4x+1;-1<x<5$

- A negative slope function, which evaluated at x equal to 5 (input), gives as output -4.

The function will be:

$y=-x+1$

Proving that it's the correct equation by evaluating "x" equal to 5, we have:

$y=-5+1$

$y=-4$

It proves that the equation is correct.

Then, for the piecewise function it will be:

$-x+1\geq 5$

Hence, we have that the piecewise function that represents the graph, is the option A (first option):

f(x) (piecewise function):

$8; x\leq -1\\\\x^{2} -4x+1;-1<x<5\\\\-x+1\geq 5$

Have a nice day!