Question # 1
Part A) Is the relation a function? Explain.
A function relates each element of a set with exactly one element of another set.
Important things for a relationship to be a function:
- Every element in X is related to some element in Y.
- A function cannot have one-to-many relationship.
- A function must contain single valued, means it is not having one-to-many relation
Considering the points on coordinate plane
(-4, 2), (-3, 0), (-2, -1), (0, 2), (2, -3), (3, 3)
If we carefully observe, we determine that relation
- relates each element of a set with exactly one element of another set
- is single-valued, means It is not giving back 2 or more results for the same input. In other words, it is not having one-to-many relation.
- It is in fact, having many to one. For example, the pairs (0, 2) and (-4, 2) is having many- to-one relationship.
So, from the above observation, it is clear that the relationship is a function.
Part B) What is the domain of the relation?
Considering the points on coordinate plane
Also we know that domain of the relation is the set of all the x-values of an ordered pairs.
So, the domain of the relation: {-4, -3, -2, 0, 2, 3}
Part C) What is the range of the relation?
Considering the points on coordinate plane
As we know that the range of a relationship is the set of all the y-values of an ordered pair.
So, the range of the relationship will be: { -3, -1, 0, 2, 3}
Note:
- The duplicated entries in the domain and range are written only once.
- Also, the domain and range can be written in ascending order.
Part D) What is the value of y when x = 2? Explain
Considering the points on coordinate plane
From the given points on the coordinate plane, it is clear that when the value of x = 2, then the value of y = -3
Therefore, the value of y is -2 when x = 2
Question # 2
Considering the points on coordinate plane
(-4, -1), (-2, 1), (0, -3), (2, 3), (4, -2)
If we bring a point, let say (2, 4), and graphed on the coordinate system, then the relation will no longer be function.
The reason is that the induction of the point (2, 4) would violate the definition of a relation to be a function.
Observe that (2, 4) and (2, 3) will make the relation having one-to-many relationship as (2, 4) and (2, 3) is giving 2 outputs i.e. y = 4, and y = 3 for a single input i.e. x = 2.
Therefore, the induction of the point (2, 4), when graphed, makes the relation not a function.
Question # 3
Part A)
[tex]f\left(x\right)\:=\:|x\:-\:3|\:-\:2[/tex]; [tex]x = -5[/tex]
The attached figure a shows the graph for the function
[tex]f\left(x\right)\:=\:|x\:-\:3|\:-\:2[/tex]
In the attached figure a, the graph represents an absolute value relationship as the absolute value of a number is never negative.
Evaluate the function for x = -5
[tex]f\left(x\right)\:=\:|x\:-\:3|\:-\:2[/tex]
[tex]|\left-5\right\:-\:3|\:-\:2....[A][/tex]
Solving
[tex]\left|-5-3\right|[/tex]
[tex]\mathrm{Subtract\:the\:numbers:}\:-5-3=-8[/tex]
[tex]=\left|-8\right|[/tex]
[tex]\mathrm{Apply\:absolute\:rule}:\quad \left|-a\right|=a[/tex]
[tex]\left|-8\right|=8[/tex]
So,
[tex]\left|-5-3\right|=8[/tex]
Equation [A] becomes
[tex]\:|-5\:-\:3|\:-\:2\:=8\:-\:2\:[/tex] ∵ [tex]\left|-5-3\right|=8[/tex]
[tex]=6[/tex]
Therefore,
the value of [tex]f\left(x\right)\:=\:|x\:-\:3|\:-\:2[/tex] at [tex]x = -5[/tex] will be 6.
i.e. [tex]f(x)=6[/tex]
Part B)
[tex]g\left(x\right)=1.5x;\:x=0.2[/tex]
The attached figure b shows the graph for the function
[tex]g\left(x\right)=1.5x[/tex]
In the attached figure b, the graph shows that the function represents a linear relationship as the graph is a straight line.
Evaluate the function for x = 0.2
As
[tex]g\left(x\right)=1.5x[/tex]
Putting x = 0.2
[tex]g\left(x\right)=1.5\left(0.2\right)[/tex]
As
[tex]1.5\left(0.2\right)=0.3[/tex]
So
[tex]g\left(x\right)=0.3[/tex]
So,
the value of [tex]g\left(x\right)=1.5x[/tex] at [tex]x = 0.2[/tex] will be 0.3.
i.e. [tex]g\left(x\right)=0.3[/tex]
Part C)
[tex]p\left(x\right)\:=\:|7\:-\:2x|;\:x\:=\:-3[/tex]
As the absolute value of a number will be never negative.
The attached figure c shows the graph for the function
[tex]p\left(x\right)\:=\:|7\:-\:2x|[/tex]
In the attached figure c, the graph represents an absolute value relationship as the absolute value of a number is never negative.
Evaluate the function for x = -3
[tex]p\left(x\right)\:=\:|7\:-\:2x|[/tex]
Solving
[tex]\left|7-2x\right|[/tex]
[tex]\left|7-2\left(-3\right)\right|[/tex]
[tex]=\left|7+6\right|[/tex]
[tex]=\left|13\right|[/tex]
[tex]\mathrm{Apply\:absolute\:rule}:\quad \left|a\right|=a,\:a\ge 0[/tex]
[tex]=13[/tex]
So,
[tex]\left|7-2x\right|=13[/tex]
So,
the value of [tex]p\left(x\right)\:=\:|7\:-\:2x|[/tex] at [tex]x = -3[/tex] will be 13.
i.e [tex]p\left(x\right)\:=13[/tex]
Keywords: function, relation
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