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DUE IN 40 MIN PLEASE HELP

DUE IN 40 MIN PLEASE HELP-example-1
User Child
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Answer:


f(x) = $\begin{cases}3x-4 &amp; -2\le x < 1\\x + 3 &amp; 1 \le x \le 5\\8 &amp; 5 \le x \le 8 \end{cases}$

Explanation:

The permissible values of x for a function is called the domain of the function. Permissible values mean that the function is real and defined.

The range of a function is the set of all y values for the domain

If a point on a graph is a filled circle, that point is included in the domain(x-value) and represents a ≤ or a ≥ inequality

If a point on a graph is an unfilled or hollow circle, that point is not included in the domain(x-value) and represents a < or a >inequality

Let's look at the three parts of the given piecewise graph

Piecewise part A

We see that the minimum and maximum values of x for this function are x = -2 and x = 1 respectively. The point corresponding to x= 1 is a hollow circle so it is not included in the domain whereas the point corresponding to x = -2 is included

Let's find the equation of this line:
Take 2 points on the line. We choose points (0, -4) and (1, -1)

Slope of this line is -1 -(-4)/(1-0) = (-1 + 4)/1 = 3
The y-intercept is -4

So equation of the line is y = 3x -4 for -2 ≤ x < 1

Proceeding in a similar manner we can determine the domain and function equation for the other two pieces, B and C

Piecewise part B

Domain of x : 1 ≤ x ≤5
Slope = (8-4)/(5-1) = 4/4 = 1

Equation is of the form y = 1x + b
To calculate b, plug in values of x = 1, y = 4 to get
4 = 1(1) + b

b = 3

Equation of this piecewise function is
y = x + 3 for 1 ≤ x ≤ 5

Piecewise Part C

This is a flat horizontal line at y = 8
So that is the equation of the line
The domain of this piece is 5 ≤ x ≤ 8
So we get y = 8 for
$\begin{cases}3x-4 &amp; -2\le x < 1\\x + 3 &amp; 1 \le x \le 5\\8 &amp; 5 \le x \le 8 \end{cases}$

Normally, instead of using y we use f(x)

Putting all this together we get the piecewise function defined as


f(x) = $\begin{cases}3x-4 &amp; -2\le x < 1\\x + 3 &amp; 1 \le x \le 5\\8 &amp; 5 \le x \le 8 \end{cases}$

User Alesko
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