Understanding the Relationship and Finding the Expression for g'(2) | A Math Explanation

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 first need to understand the relationship between the function f and its inverse function g

To find the expression for g'(2), we first need to understand the relationship between the function f and its inverse function g.

The function f and its derivative are given in a table. Let’s assume the table looks like this:

| x | f(x) | f'(x) |
|——-|———-|——–|
| 1 | 5 | 2 |
| 2 | 4 | 3 |
| 3 | 3 | 2 |
| 4 | 2 | 1 |
| 5 | 1 | 2 |

Now, let’s focus on the function g(x) = f^(-1)(x), which is the inverse function of f. The inverse function swaps the x and y values of f, meaning that if (a, b) is a point on f, then (b, a) is a point on g.

So, to find the value of g'(2), we want to find the derivative of the inverse function g at x = 2.

To do this, we can use the fact that the inverse function has the property that the derivative of the inverse function at a point is equal to 1 divided by the derivative of the original function at the corresponding point.

In equation form:

g'(x) = 1 / f'(f^(-1)(x))

Now, let’s find the corresponding point on f for which x = 2. By looking at the table, we see that when x = 2, f(x) = 4.

So, we have:

g'(2) = 1 / f'(f^(-1)(2))

g'(2) = 1 / f'(4)

Now, let’s find the value of f'(4) using the table. Looking at the table, we see that when x = 4, f'(x) = 1.

Therefore:

g'(2) = 1 / 1

Finally, simplifying:

g'(2) = 1

So, the correct expression for g'(2) is simply 1.

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