G(x)= 3 - |2x-4| is defined for the domain -1 < x < 6.
1 answer:
(a) The range of g is the set of y values that g covers for the given domain. Given the shape of g, you will be interested in the starting point and ending point g(-1)=1, g(6)=-5 and the tip of g, which is where |2x-4|=0, ie., x=2. g(2) =3.
The highest encountered g is 3 (at x=2). The lowest one is -5 (at x=6). Note that the highest is inclusive, the lowest is exclusive (since x=6 is not part of the domain).
So the range is (-5,3].
(b) Solve g(x) = 2, then find the x values that cause g(x)>2.
g(x)=2 => 3-|2x-4| = 2 => |2x - 4| = 1 => | x-2 | = 1/2
x-2 = 1/2 or 2-x = 1/2 => x=5/2 or x=3/2
From the graph you can see that x has to be in between these values, so the solution is:
3/2 < x < 5/2.
The values themselves are excluded (ie., no ≤) since g(x) > 2 and not ≥ 2
The highest encountered g is 3 (at x=2). The lowest one is -5 (at x=6). Note that the highest is inclusive, the lowest is exclusive (since x=6 is not part of the domain).
So the range is (-5,3].
(b) Solve g(x) = 2, then find the x values that cause g(x)>2.
g(x)=2 => 3-|2x-4| = 2 => |2x - 4| = 1 => | x-2 | = 1/2
x-2 = 1/2 or 2-x = 1/2 => x=5/2 or x=3/2
From the graph you can see that x has to be in between these values, so the solution is:
3/2 < x < 5/2.
The values themselves are excluded (ie., no ≤) since g(x) > 2 and not ≥ 2
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Answer:
Problem 2):
which agrees with answer C listed.
Problem 3) : D = (-3, 6] and R = [-5, 7]
which agrees with answer D listed
Step-by-step explanation:
Problem 2)
The Domain is the set of real numbers in which the function (given by a graph in this case) is defined. We see from the graph that the line is defined for all x values between 0 and 240. Such set, expressed in "set builder notation" is:
Problem 3)
notice that the function contains information on the end points to specify which end-point should be included and which one should not. The one on the left (for x = -3 is an open dot, indicating that it should not be included in the function's definition, therefor the Domain starts at values of x strictly larger than -3. So we use the "parenthesis" delimiter in the interval notation for this end-point. On the other hand, the end point on the right is a solid dot, indicating that it should be included in the function's definition, then we use the "square bracket notation for that end-point when writing the Domain set in interval notation:
Domain = (-3, 6]
For the Range (the set of all those y-values connected to points in the Domain) we use the interval notation form:
Range = [-5, 7]
since there minimum y-value observed for the function is at -5 , and the maximum is at 7, with a continuum in between.