Find b & Inverse Function of g(x) = 1-x2

In summary, to find the smallest real value b such that g has an inverse function, we can graph g(x) = 1-x2 with the restriction g:[b,2] -> R and look for the portion of the graph that makes g(x) one-to-one. The answer is b=0.
  • #1
t_n_p
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Homework Statement



Let g:[b,2] -> R where g(x) = 1-x2. If b is the smallest real value such that g has an inverse function, find b and g inverse

The Attempt at a Solution



I can find the inverse function easily, but I don't understand how I go about finding b.

According to the book, answer is b=0
 
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  • #2
I would just look at the graph. Start by graphing g(x) = 1-x2 with the restriction g:[-∞,2] -> R, and then look to see how much of the right portion of the graph is needed to make g(x) one-to-one.
 
  • #3
right that makes sense!

In other words you can only take the inverse of a function f, if the original function f is one-to-one.

got it.
 

1. What is the value of b in the equation g(x) = 1-x2?

The value of b in this equation is 1.

2. How do you find the inverse function of g(x) = 1-x2?

To find the inverse function of g(x), we first replace g(x) with y. This gives us the equation y = 1-x2. Next, we switch the x and y variables and solve for y. This results in the inverse function g^-1(x) = √(1-x).

3. Can the value of b in g(x) = 1-x2 be negative?

Yes, the value of b can be negative. This would result in a reflected parabola with its vertex at the point (0,-b).

4. What is the domain of g(x) = 1-x2?

The domain of g(x) is all real numbers since there are no restrictions on the x variable in the equation.

5. What is the range of g(x) = 1-x2?

The range of g(x) is all real numbers greater than or equal to -1, since the highest value the function can reach is 1, and the lowest value is -∞.

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