Questions: For the linear function y=f(x)=-5x+8: a. Find d f/d x at x=-5. b. Find a formula for x=f^(-1)(y).

Transcript text: For the linear function $y=f(x)=-5 x+8$ : a. Find $\frac{d f}{d x}$ at $x=-5$. b. Find a formula for $x=f^{-1}(y)$.

Solution

Solution Steps

To solve the given problems, we need to follow these steps:

a. The derivative of a linear function $ y = f(x) = -5x + 8 $ is simply the coefficient of $ x $, which is the slope of the line. Therefore, $ \frac{d f}{d x} = -5 $. Since the derivative is constant, it is the same at any point, including $ x = -5 $.

b. To find the inverse function $ x = f^{-1}(y) $, we need to solve the equation $ y = -5x + 8 $ for $ x $. This involves isolating $ x $ on one side of the equation.

Step 1: Derivative of the Function

The given linear function is $ y = f(x) = -5x + 8 $. The derivative of a linear function is the coefficient of $ x $, which represents the slope of the line. Therefore, the derivative is:

\[ \frac{d f}{d x} = -5 \]

Since the derivative is constant, it is the same at any point, including $ x = -5 $.

Step 2: Inverse of the Function

To find the inverse function $ x = f^{-1}(y) $, we start with the equation of the line:

\[ y = -5x + 8 \]

We solve for $ x $ by isolating it on one side of the equation:

\[ y - 8 = -5x \]

\[ x = \frac{y - 8}{-5} \]

Thus, the inverse function is:

\[ f^{-1}(y) = \frac{y - 8}{-5} \]