Convert log6 x = 10 to an exponential equation
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(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