Find an exponential function g(x) = b^x such that the graph of g passes through

Question

Find an exponential function g(x) = b^x such that the graph of g passes through the given point. (3, 216) (-1, e^2) (-1, 5) (2, e) Consider f(x) = -5 - x/3 - 2x. Find f^-1 (7) without finding f^-1(x) Find f^-1 (x) Does the function g(x) = 2x^2 - 9 have an inverse? If it does find it. If it does not then explain why this is so. Does the function p(x) = 3 Squareroot 2x + 1 have an inverse? If it does find it. If it does not then explain why this is so. Recall the function from 3(a) above. Restrict the domain of g such that the function has an inverse. Explain why you chose the domain that you did. Is this unique? How do you know? Find g^-1 using the domain restriction you stated in (a). Is this unique? Explain your g answer.

Explanation / Answer

f(x) = (-5 -x)/(3 - 2x)

a) f^-1(7) can be found without finding inverse by solving for x:

7 = ( -5 -x)/( 3 -2x)

21 - 14x = -5 -x

-13x = -26 ; x = 2

So, f^-1(7) = 2

b) f^-1(x)

y = f(x)

Plug x= y and y = x and solve for y

y = (-5 -x)/(3 - 2x)

x = ( -5 - y)/( 3 -2y)

3x - 2xy = -5 -y

y(1 -2x) = -5 -3x

y = ( -5 -3x)/( 1- 2x)

f^-1(x) = ( -5 -3x)/( 1- 2x)

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