Question

Civen \(y=7 x+21\), find its inverse function. Then find \(d y / d x\) and \(d x / d y\), and verify the inverse-function rule. Also verify that the graphs of the two functions bear a mirrorimage relationship to each other.

Short answer

The inverse of \( y = 7x + 21 \) is \( y = \frac{x - 21}{7} \). \( dy/dx = 7 \) and \( dx/dy = 1/7 \), verifying the inverse-function rule. The graphs are mirror images over \( y = x \).

Step-by-step solution

Understand the Inverse Function

To find the inverse function of a given function, we need to reverse the roles of the dependent and independent variables. For the given function \( y = 7x + 21 \), the role of \( y \) and \( x \) will be exchanged to determine the inverse function.

Solve for the Inverse Function

Starting from \( y = 7x + 21 \), replace \( y \) with \( x \) and \( x \) with \( y \) to get \( x = 7y + 21 \). Solving for \( y \), we get \( y = \frac{x - 21}{7} \). Therefore, the inverse function is \( y = \frac{x - 21}{7} \).

Calculate the Derivative dy/dx

The derivative of \( y = 7x + 21 \) with respect to \( x \) is \( \frac{dy}{dx} = 7 \). This derivative represents the rate of change of \( y \) with respect to \( x \).

Calculate the Derivative dx/dy

For the inverse function \( y = \frac{x - 21}{7} \), compute the derivative of \( x \) with respect to \( y \), which is \( \frac{dx}{dy} = 7 \) because solving \( y = \frac{x - 21}{7} \) gives \( x = 7y + 21 \).

Verify the Inverse Function Rule

According to the inverse function rule, \( \frac{dy}{dx} \cdot \frac{dx}{dy} = 1 \). Substitute \( \frac{dy}{dx} = 7 \) and \( \frac{dx}{dy} = 7 \), so \( 7 \times \frac{1}{7} = 1 \). This verifies the inverse function rule.

Confirm Mirror Image Relationship

The graphs of \( y = 7x + 21 \) and its inverse will be reflections of each other across the line \( y = x \). Plotting these lines will show that each point on one graph can be reflected over \( y = x \) to coincide with a point on the other graph.

Key concepts

Derivative

The concept of a derivative is fundamental in calculus. A derivative represents the rate at which a function is changing at any given point.
For any function, if you think about the graph, the derivative tells us how steep the graph is at a particular point.
If the derivative is positive, the function is increasing; if negative, it's decreasing.
In the case of the function given, i.e., \[ y = 7x + 21 \]
the derivative is computed with respect to the variable x:

• Given that the derivative of \( 7x \) is 7.

This means, for every unit increase in x, y increases by 7 units.
This is vital in understanding how quickly or slowly changes occur in the function.

Inverse Function Rule

The inverse function is like the reverse gear of a function.
It takes the output of the original function and traces back to find the input.
To find an inverse, we swap the dependent variable with the independent variable and then solve for the new dependent variable.For the function \( y = 7x + 21 \):

• Switch x with y, giving \( x = 7y + 21 \).
• Solve for y to find the inverse: \( y = \frac{x - 21}{7} \).

Once we have both functions, the inverse function rule comes into play.
This rule states that if you multiply the derivative of the original function with the derivative of the inverse function, the result should be 1:

• \( \frac{dy}{dx} \times \frac{dx}{dy} = 1 \)

In our example:

• \( \frac{dy}{dx} = 7 \) for the original function.
• \( \frac{dx}{dy} = \frac{1}{7} \) for the inverse function.

Since \( 7 \times \frac{1}{7} = 1 \), this correctly verifies the rule.

Mirror Image Relationship

The mirror image relationship is a fascinating concept tied to functions and their inverses.
Imagine looking in a mirror: every action has an equal and opposite reflection.
This is the same for the graphs of a function and its inverse.For the function \( y = 7x + 21 \) and its inverse \( y = \frac{x - 21}{7} \),

• Their graphs are reflections across the line \( y = x \).

This means every point on one graph corresponds directly to a point on the other, mirrored perfectly.
In practical terms, if you graph these functions, you will see this reflection symmetry, showing a robust visual confirmation of their inverse nature.

Graph Reflection

Graph reflection is not just about drawing pretty pictures; it helps us understand deeper mathematical truths.
Reflecting a graph over a line shows symmetry, one of mathematics' core beauties. When you reflect a graph across the line \( y = x \):

• Each point \( (a, b) \) on the original function becomes \( (b, a) \) on the inverse.

For our function \( y = 7x + 21 \):

• Plotting gives every x, y turned around on the inverse graph.

Seeing this helps solidify understanding that inverse functions are two sides of the same coin.
Noticing how they reflect across \( y = x \) visually reinforces the mathematical concepts behind derivatives, inverse rules, and mirror images.