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Question 1123544: I've been trying to solve this problem for a while and would appreciate the help thank-you.
The one-to-one function g and h are defined as:
g(x)=x-9/5
h={(-7,-1),(-4,4),(-1,3),(7,1)}
Find:
g^-1(x) =
(g ∘ g^-1)(-2) =
h^-1(-1) =
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! g(x) = x - 9/5
let y = g(x)
you get y = x - 9/5
replace y with x and x with y to get:
x = y - 9/5
solve for y in terms to x to get:
y = x + 9/5
replace y with g^-1(x)
you have g(x) = x - 9/5 and g^-1(x) = x + 9/5
g^-1(x) is the inverse function of g(x).
put them on a graph and you will see that they are reflections about the x-axis.
this means that the point (x,y) in g(x) will be directly opposite the point (y,x) in g^-1(x) and equidistant from the line y = x.
here's my graph, using the point (3.4) in g(x) which is reflected about the line y = x to become the point (1.6,3.4) in g^-1(x).
g(x) = x - 9/5.
when x = 3.4, g(x) becomes 3.4 - 9/5 = 1.6
g^-1(x) = x + 9/5.
when x = 1.6, g^-1(x) becomes 1.6 + 9/5 = 3.4
the point (3.4,1.6) is a reflection of the point (1.6,3.4) about the line y = x.
this means the the point (3.4,1.6) is the same vertical distance from the line y = x as the point (1.6,3.4).
the dashed line y = -x was used to create this vertical connection through the dashed line y = x.
here's a reference on inverse functions.
https://www.mathsisfun.com/sets/function-inverse.html
the other property of an inverse function to note is that:
g(g^-1(x)) = x and g^-1(g(x)) = x
g(x) = x - 9/5
g^-1(x) = x + 9/5
g(g^-1(x)) = g(x + 9/5) = (x + 9/5) - 9/5 = x
g^-1(g(x)) = g^-1(x - 9/5) = (x - 9/5) + 9/5 = x
the function gets you somewhere.
the inverse function gets you back.
for example if f = (1,2), the f^-1) = (2,1)
the function f gives you y = 2 when x = 1.
the function f^-1) gives you y = 1 when x = 2.
there are restrictions as noted in the reference.
to answer your questions:
g(x)=x-9/5
h={(-7,-1),(-4,4),(-1,3),(7,1)}
find g^-1(x):
we did that above.
set y = x - 9/5
replace y with x and x with y to get x = y - 9/5
solve for y to get y = x + 9/5
g^-1(x) = x + 9/5
find (g ∘ g^-1)(-2):
g(x) = x - 9/5
g^-1(x) = x + 9/5
(g ∘ g^-1)(x) = g(g^-1(x))
g(x) = x - 9/5
g^-1(x) = x + 9/5
g(g^-1(x)) = g(x + 9/5) = (x + 9/5) - 9/5 = x
you replace x in g(x) = x - 9/5 with g^-1(x) = x + 9/5 to get g(x + 9/5) = (x + 9/5) - 9/5 = x
this is as it should be since, by definition:
g(g^-1(x)) = x and g^-1(g(x)) = x
the problem says find (g ∘ g^-1)(-2):
g^-1(-2) = (-2) + 9/5 = -2 + 9/5 = -10/5 + 9/5 = -1/5
g(g^-1(-2)) becomes g(-1/5) which becomes (-1/5) - 9/5 which becomes -10/5 which becomes -2.
therefore, (g ∘ g^-1)(-2) = -2
here's a review on functions.
https://www.mathsisfun.com/sets/function.html
here's a review on functional notation.
https://mathbitsnotebook.com/Algebra1/Functions/FNNotationEvaluation.html
find h^-1(-1):
you are given that:
h={(-7,-1),(-4,4),(-1,3),(7,1)}
when x = -7, y = -1
when x = -4, y = 4
when x = -1, y = 3
when x = 7, y = 1
h^-1 is the inverse function of h.
therefore, in the inverse function, .....
when x = -1, y = -7
when x = 4, y = -4
when x = 3, y = -1
when x = 1, y = 7
h = {(-7,-1),(-4,4),(-1,3),(7,1)} becomes:
h^-1 = {(-1,-7),(4,-4),(3,-1),(1,7)}
this agrees with the definition that, the point (x,y) in h(x) becomes the point (y,x) in h^-1(x).
that is the point that is directly opposite and equidistant from the line y = x.
your y value in the regular function becomes your x value in the inverse function and your x value in the regular function becomes your y value in the inverse function.
that's why h^-1(-1) = -7.
h went from -1 to -7
h^-1 goes from -7 to -1.
i think i'm right, but i could be wrong, so, if you find something confusing to you that doesn't make any sense, email me at dtheophilis@gmail.com and i'll try to explain further and / or correct any errors as best i can.
read the references.
they should help.
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