Question

Given ( k(x) = 2x - 7/3 ), find ( k(-4) ). Next, find ( k⁻¹(9) ) in two ways: first by finding ( k⁻¹(x) ) and then finding ( k⁻¹(9) ), and second by using a property of functions and inverses to find ( k⁻¹(9) ) from ( k(x) ).

Answer 1

To find k(-4), plug -4 into the expression for k(x). To find k⁻¹(9), either find the inverse function by switching the roles of x and y, or plug 9 into k(x) and solve for x.

Step-by-step explanation:

To find k(-4), plug -4 into the expression for k(x).

So, k(-4) = 2(-4) - \frac{7}{3}.

Simplifying, we get k(-4) = -8 - \frac{7}{3}. To find k⁻¹(9), we need to find the inverse function of k(x).

The inverse function is found by switching the roles of x and y.

So, let y = 9 and solve for x in the equation 9 = 2x - \frac{7}{3}.

By solving for x, we find that k⁻¹(9) = \frac{8}{3}.

Alternatively, using the property of functions and inverses, we can find k⁻¹(9) from k(x) by plugging 9 into k(x) and solving for x. So, 9 = 2x - \frac{7}{3}.

Rearranging, we find that x = \frac{8}{3}.