The table above gives selected values for a differentiable and decreasing function f and its derivative. If g(x)=f−1(x) for all x, which of the following is a correct expression for g′(2) ?
To find the expression for g'(2), we need to understand the relationship between the function f and its inverse function g
To find the expression for g'(2), we need to understand the relationship between the function f and its inverse function g.
Since g(x) is defined as the inverse of f(x), it means that g(f(x)) = x. In simpler terms, g undoes what f does, so when we plug in f(x) into g(x), we get back x.
To find g'(2), we first need to determine the value of f(2). Looking at the table, we see that f(2) = 4.
Now, let’s find the value of g'(2) using the relationship between f and g. We know that g(f(2)) = 2.
Differentiating both sides of this equation, we get:
g'(f(2)) * f'(2) = 1.
Since f(2) = 4, we can replace f(2) in the above equation to obtain:
g'(4) * f'(2) = 1.
We are given that f'(2) = -3, so substituting this value, we get:
g'(4) * (-3) = 1.
To find g'(4), we can rearrange the equation:
g'(4) = 1 / (-3) = -1/3.
Therefore, the correct expression for g'(2) is -1/3.
In summary, the correct expression for g'(2) is -1/3.
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