Answer:
4. Functions: 1, 2, 4, 5, 7, 8, 11, 12
5. y = -1/2x+5/2, x ≤ -1; 2x +1, x > -1.
y = 2, x < 0; x, 0 ≤ x ≤ 3; 3 x > 3.
Step-by-step explanation:
4. A list of ordered pairs is a function of no x-values are re-used. In (3) and (6), the value x=2 is used more than once.
A graph represents a function if it passes the "vertical line test." A vertical line cannot intersect the graph in 2 or more points. (9) and (10) both fail that test.
If the relation is not listed here as being "not a function," then it is one of the answers to question 4.
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5. The first step in writing the equation of a piecewise function is to identify the pieces. These are generally bounded by points of discontinuity--jumps in the function value, or changes in the slope of lines.
The second step is to identify the section of the function that boundary points belong to. Solid dots are part of the function definition; open circles are not.
<u>Left Graph</u>
The left piece ends at x=-1. There is a solid dot attached to the left piece, so its definition will be for the domain x ≤ -1. That line has slope -1/2, since is drops 1 unit for each 2 to the right. If extended, it would intersect the y-axis at y = 2 1/2 = 5/2. So, the piece on the left is y = -1/2x + 5/2 for x ≤ -1.
The right piece starts at x=-1, but does not include that point. It has a rise of 2 for each 1 to the right, so its slope is 2. It crosses the y-axis at y=1, so the piece on the right is y = 2x + 1 for x > -1.
You can use the method of your textbook author to combine theses pieces into one equation. The method shown above is one way to do it.
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<u>Right Graph</u>
This graph has a discontinuity at x=0 and a change in slope at x=3. It can be described by 3 pieces. The point at x=0 does not belong to the left piece, but goes with the middle piece.
The left piece of the function is the constant 2, so has the equation y = 2 for x < 0.
The middle piece has a slope of 1 and a y-intercept of 0, so has the equation y = x for 0 ≤ x ≤ 3.
The point at x=3 belongs to both the middle piece and the right piece, so can be part of both function definitions, if you like. Generally, it is better form to include any given x-value in only one of the pieces of the function. So the equation for the right piece can be y = 3 for x > 3.
![\frac{1}{ \frac{343}{3} } \int\limits^7_0 \int\limits^{x^2}_0 {4x\sin(y)} \, dydx = -\frac{3}{343} \int\limits^7_0 {4x\cos(x^2)} \, dx \\ \\ =-\frac{3}{343} \int\limits^{49}_0 {2\cos(t)} \, dt=-\frac{6}{343} \left[\sin(t)\right]_0^{49} \, dt=-\frac{6}{343}\sin49](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B%20%5Cfrac%7B343%7D%7B3%7D%20%7D%20%20%5Cint%5Climits%5E7_0%20%20%5Cint%5Climits%5E%7Bx%5E2%7D_0%20%7B4x%5Csin%28y%29%7D%20%5C%2C%20dydx%20%20%3D%20-%5Cfrac%7B3%7D%7B343%7D%20%20%5Cint%5Climits%5E7_0%20%7B4x%5Ccos%28x%5E2%29%7D%20%5C%2C%20dx%20%20%5C%5C%20%20%5C%5C%20%3D-%5Cfrac%7B3%7D%7B343%7D%20%5Cint%5Climits%5E%7B49%7D_0%20%7B2%5Ccos%28t%29%7D%20%5C%2C%20dt%3D-%5Cfrac%7B6%7D%7B343%7D%20%5Cleft%5B%5Csin%28t%29%5Cright%5D_0%5E%7B49%7D%20%5C%2C%20dt%3D-%5Cfrac%7B6%7D%7B343%7D%5Csin49)