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Exercises 1 For a relation to be a function, each x-value can only be paired with one y-value, but one y-value can be paired with multiple x-values. If we examine the table, we do not see any x-value with different y-values.Input, xOutput, y -10 01 14 24 38 Although the y-value 4 is paired with multiple x-values (1 and 2), the relation is a function because there are not x-values paired with multiple y-values.
Exercises 2 For a relation to be a function, each x-value can only be paired with one y-value. However, one y-value can be paired with multiple x-values. (-10,2),(-8,3),(-6,5),(-8,8),(-10,6) Examining the given points, we can see that two sets of pairs have the same x-values. (-10,2),(-10,6) and (-8,3),(-8,8) However, each set of pairs has different y-values. Therefore, the relation is not a function.
Exercises 3 To find the x-coordinate of a point, we move vertically until we hit the x-axis. Similarly, to find the y-coordinate, we move horizontally until we hit the y-axis.Now let's list the ordered pairs that we found. (0,-5),(1,-3),(2,-1),(3,1),(4,-3)​ The domain of a function is found by listing the relation's x-values. The range is found by listing the relation's y-values. Domain: Range: ​{0,1,2,3,4}{-5,-3,-1,1}​
Exercises 4 The arrows on the graph indicate that the function will continue on forever in the downward direction. This also tells us that there are no gaps in the function, and that we can input any real number for x.Therefore, we can write our domain as an interval that contains all real numbers. Domain: -∞<x<∞​ The corresponding y-coordinates, on the other hand, have an upper bound at 3 because the graph does not exist above this point on the y-axis. Therefore, the range of the function is all possible y-values less than or equal to 3. Range: y≤3​
Exercises 5 To begin, let's take a look at the given graph.The domain of a function is the set of all possible x-values. The range is the set of all possible y-values. Notice that, horizontally, our graph spans from x=-3 to x=3, inclusive. These values are the lower and upper boundaries of our domain. Domain: -3≤x≤3​ Meanwhile, vertically, our graph spans from y=-1 to y=3, inclusive. These values are the lower and upper boundaries of our range. Range: -1≤y≤3​
Exercises 6 The graph of a linear function is portrayed as a single, straight line in a coordinate plane. One way to determine if the given graph is linear is by comparing it with a straight edge, such as a ruler.We can see that all parts of the given graph lie on the same straight line. Because of this, we can conclude that the function is linear.
Exercises 7 The graph of a linear function can be portrayed as a single, straight line in a coordinate plane. To begin determining if the data given in the table represents a linear function, let's first plot the data as (x,y) coordinate pairs.If the function is linear, connecting these points will form a straight line. Otherwise, we will have shown that the function is nonlinear. Let's connect the first two points with a straight edge and observe the result.We can see that the third point does not lie on the same line in the coordinate plane. Therefore, the function is nonlinear.
Exercises 8 To determine if the function is linear, we will create a table of values.xx(2−x)y 11(2−1)1 22(2−2)0 33(2−3)-3 44(2−4)-8 Analyzing the x- and y-columns, we can see that as x increases by 1, the amount that y changes by is not constant. 1 ⟶−1​ 0 ⟶−3​ -3 ⟶−5​ -8​ Since the rate of change is not constant, the function is nonlinear. To verify, let's plot our points in a graph.We can see that, when graphed, the points do not all lie on one straight line.
Exercises 9 Before we look at the given table, let's recall the definitions of discrete and continuous.Discrete Domain: A set of input values that consist of only certain numbers in an interval. Continuous Domain: A set of input values that consist of all possible numbers in an interval.Here, the input is the depth of the sea. This is a measure of length. Even though we are only given the pressure at specific depths, the pressure is not only changing at those lengths. The pressure is not magically changing in steps, it is gradually changing over the interval as a diver is diving deeper and deeper. Therefore, the domain is continuous.
Exercises 10 Before we look at the given table, let's recall the definitions of discrete and continuous.Discrete Domain: A set of input values that consist of only certain numbers in an interval. Continuous Domain: A set of input values that consist of all possible numbers in an interval.In the given situation, the input is the number of hats. It is only possible to have whole number values for the "number of hats." We can have 2 hats, 3 hats, or 4 hats, but not 3.14159 hats. Therefore, the domain is discrete.
Exercises 11 To find the value of x that will make the function equal to -3, we will substitute -3 for w(x) into the given function rule. Then we can solve for x. w(x)=-2x+7w(x)=-3-3=-2x+7LHS−7=RHS−7-10=-2xLHS/(-2)=RHS/(-2)5=xRearrange equationx=5 When x=5, the function equals -3. ⧼ebox-type-answer-check⧽ info Verifying Our Solution We can verify our solution by substituting x=5 into the rule and evaluating. w(x)=-2x+7x=5w(5)=-2(5)+7Multiplyw(5)=-10+7Add termsw(5)=-3 When x=5, w(x)=-3, so our solution is correct.
Exercises 12 To graph using a table, we first select some arbitrary values from the domain and substitute them for x in the given equation. Simplifying the resulting expression for g(x) will give us the corresponding values from the range.xg(x)=x+3Simplify 0g(0)=0+3g(0)=3 2g(2)=2+3g(2)=5 4g(4)=4+3g(4)=7 Using these x- and g(x)-values as (x,g(x)) coordinate pairs, we can plot the points. Connecting these points will give us the graph of the function.
Exercises 13 To graph using a table, we first select some arbitrary values from the domain and substitute them for x in the given equation. Simplifying the resulting expression for p(x) will give us the corresponding values from the range.xp(x)=-3x−1Simplify 0p(0)=-3(0)−1p(0)=-1 2p(2)=-3(2)−1p(2)=-7 4p(4)=-3(4)−1p(4)=-13 Using these x- and p(x)-values as (x,p(x)) coordinate pairs, we can plot the points. Connecting these points will give us the graph of the equation.
Exercises 14 To graph using a table, we first select some arbitrary values from the domain and substitute them for x in the given equation. Simplifying the resulting expression for m(x) will give us the corresponding values from the range.xm(x)=32​xSimplify 0m(0)=32​(0)m(0)=0 3m(3)=32​(3)m(3)=2 6m(6)=32​(6)m(6)=4 Using these x- and m(x)-values as (x,m(x)) coordinate pairs, we can plot the points. Connecting these points will give us the graph of the equation.
Exercises 15
Exercises 16