g(x) = 2(x + 3)^2 − 8 (−3 ≤ x ≤ 0).?
(i.)Specify the domain and image set of the inverse function g^−1, and find its rule.
(ii) Sketch the graph of y = g^−1(x).
pls explain me i am really confused for both queries
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2(x + 3)^2 − 8=2(x^2+6x+9)-8
=2x^2+12x+18-8
=2x^2+12x+10
=2(x+1)(x+5)
x should be between....(-1 ≤ x ≤ -5)and(−3 ≤ x ≤ 0)
therefore the answer is (-3 ≤ x ≤ -1)
the inverse function is -2x^2-12x-10
graph is going across (0,-1) and (0,-5)
the minimum value of the graph is g(x)=10
hello Maria,
when finding inverse function, one usually swaps variables (y becomes x and x becomes y) and then solve for
y= whatever
start:
y = 2(x + 3)^2 â 8
swap variables:
x=2(y+3)^2-8
solve for y:
x+8 = 2(y+3)^2
(x+8)/2 = (y+3)^2
swap sides for readability
(y+3)^2=(x+8)/2
now take a root (so we can get rid of square on the left side):
y+3= [(x+8)/2] ^ 0.5
y= [(x+8)/2] ^ 0.5 - 3
this is inverse function of g(x)
to see the sketch, just type the equation into wolframalpha.
note that if you draw a symmetry line y=x, it will act as a mirror between g(x) and inverse of g(x)