Question

In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using \(y=f(x)\) to represent a function, an applied problem might use \(C=C(q)\) to represent the cost C of manufacturing qunits of a good. Because of this, the inverse notation \(f^{-1}\) used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as \(C=C(q)\) will be \(q=q(C) .\) So \(C=C(q)\) is a function that represents the cost \(\mathrm{C}\) as a function of the number \(q\) of units manufactured, and \(q=q(C)\) is a function that represents the number \(q\) as a function of the cost \(C\). Illustrate this idea. Height and Head Circumference The head circumference \(C\) of a child is related to the height \(H\) of the child (both in inches) through the function $$ H(C)=2.15 C-10.53 $$ (a) Express the head circumference \(C\) as a function of height \(H\) (b) Verify that \(C=C(H)\) is the inverse of \(H=H(C)\) by showing that \(H(C(H))=H\) and \(C(H(C))=C\) (c) Predict the head circumference of a child who is 26 inches tall.

Short answer

The function is \(C(H) = \frac{H + 10.53}{2.15}\). Verified, and for 26 inches tall, the head circumference is approximately 16.99 inches.

Step-by-step solution

Express head circumference as a function of height

Start with the given function \(H(C) = 2.15C - 10.53\). To express the head circumference \(C\) as a function of height \(H\), solve for \(C\): 1. Set up the equation: \(H = 2.15C - 10.53\). 2. Add 10.53 to both sides: \(H + 10.53 = 2.15C\). 3. Divide both sides by 2.15: \(C = \frac{H + 10.53}{2.15}\). Thus, the function is \(C(H) = \frac{H + 10.53}{2.15}\).

Verify the inverse functions

Verify that \(H(C(H)) = H\): 1. Substitute \(C(H)\) into \(H(C)\): \(H(C(H)) = H\left( \frac{H + 10.53}{2.15} \right)\). 2. Substitute the expression in the function \(H(C)\): \(2.15 \left( \frac{H + 10.53}{2.15} \right) - 10.53\). 3. Simplify: \(H + 10.53 - 10.53 = H\). 4. That's verified: \(H(C(H)) = H\). 5. Now verify \(C(H(C)) = C\): Substitute \(H(C)\) into \(C(H)\): \(C(H(C)) = \frac{H(C) + 10.53}{2.15}\). 6. Substitute \(H(C)\) in the function \(2.15C - 10.53\): \(\frac{2.15C - 10.53 + 10.53}{2.15}\). 7. Simplify: \(\frac{2.15C}{2.15} = C\). 8. That's verified: \(C(H(C)) = C\).

Predict the head circumference of a 26-inch tall child

Use the inverse function \(C(H)\) to find the circumference when \(H = 26\): 1. Plug in 26 for \(H\): \(C(26) = \frac{26 + 10.53}{2.15}\). 2. Calculate the numerator: \(26 + 10.53 = 36.53\). 3. Divide by the coefficient: \(\frac{36.53}{2.15} \approx 16.99\). Therefore, the predicted head circumference is approximately 16.99 inches.

Key concepts

Independent Variable

In any function, the independent variable is the input variable that we control. In the context of the given problem, the height of a child, represented by the letter \(H\), acts as the independent variable. This means we can choose different values for height, and the function will output the corresponding value for head circumference. Think of the independent variable as the cause in a cause-and-effect relationship. The input we provide here is the child's height.

Dependent Variable

The dependent variable is the output of the function, which depends on the value of the independent variable. In this scenario, the head circumference, represented by \(C\), is the dependent variable. When you input a height into the function, it calculates the head circumference based on that height. Therefore, the head circumference is dependent on the child's height. Imagine it like the effect in a cause-and-effect relationship—the circumference we get depends on our input height.

Function Notation

Function notation is a way to clearly and concisely write functions and their relationships. Instead of using \(y\) and \(x\), in applied problems, we use letters that represent quantities in real-world situations. For instance, we write \(H(C)=2.15C-10.53\) to show how height \(H\) is dependent on head circumference \(C\). Function notation simplifies complex relationships by expressing them in a manageable form. It also makes it easier to track the independent and dependent variables.

Solving Equations

Solving equations involves manipulating an equation to isolate the desired variable. In this case, we started with the equation \(H=2.15C-10.53\) and solved for \(C\) to express \(C\) as a function of \(H\). Here’s how:
1. Add \(10.53\) to both sides of the equation: \(H + 10.53 = 2.15C\).
2. Divide both sides by \(2.15\): \(C = \frac{H + 10.53}{2.15}\).
The resultant equation \(C(H) = \frac{H + 10.53}{2.15}\) is the inverse, showing \(C\) as a function of \(H\).

Verification of Inverses

To verify that two functions are inverses, we must show that one function undoes the other. For \(H(C)\) and \(C(H)\), we need to ensure \(H(C(H))=H\) and \(C(H(C))=C\). Here's the step-by-step:
1. Substitute \(C(H)\) into \(H(C)\): \(H(C(H)) = H\bigg(\frac{H + 10.53}{2.15}\bigg)\). Simplify to \(H(C(H)) = H\).
2. Substitute \(H(C)\) into \(C(H)\): \(C(H(C)) = \frac{H(C) + 10.53}{2.15}\). Simplify to \(C(H(C)) = C\).
This verification ensures that inputting a value and then applying its inverse returns the original value, confirming they are indeed inverse functions.