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. Ideal Body Weight One model for the ideal body weight \(W\) for men (in kilograms) as a function of height \(h\) (in inches) is given by the function $$ W(h)=50+2.3(h-60) $$ (a) What is the ideal weight of a 6 -foot male? (b) Express the height \(h\) as a function of weight \(W\). (c) Verify that \(h=h(W)\) is the inverse of \(W=W(h)\) by showing that \(h(W(h))=h\) and \(W(h(W))=W\) (d) What is the height of a male who is at his ideal weight of 80 kilograms?

Short answer

The ideal weight of a 6-foot male is 77.6 kg. The height function is h = (W - 50)/2.3 + 60. h(W(h)) = h, W(h(W)) = W. A male who is 80 kg is 73.04 inches tall.

Step-by-step solution

Convert height from feet to inches

To find the ideal weight of a 6-foot male, first convert the height to inches. 1 foot = 12 inches, so 6 feet = 6 × 12 = 72 inches.

Use the weight function

The weight function given is:\[ W(h) = 50 + 2.3 (h - 60) \]Substitute the height in inches into the function:\[ W(72) = 50 + 2.3 (72 - 60) \]

Simplify the expression

Calculate the value inside parentheses:\[ 72 - 60 = 12 \]Multiply the result by 2.3:\[ 2.3 \times 12 = 27.6 \]Add the values to find W(72):\[ W(72) = 50 + 27.6 = 77.6 \text{ kg} \]So the ideal weight of a 6-foot male is 77.6 kg.

Express height h as a function of weight W

Starting from the weight function: \[ W(h) = 50 + 2.3 (h - 60) \]Isolate h by solving for h in terms of W.First, subtract 50 from both sides:\[ W - 50 = 2.3 (h - 60) \]Next, divide both sides by 2.3:\[ \frac{W - 50}{2.3} = h - 60 \]Finally, add 60 to both sides:\[ h = \frac{W - 50}{2.3} + 60 \]Thus, the function is:\[ h = h(W) = \frac{W - 50}{2.3} + 60 \]

Verify h(W(h)) = h and W(h(W)) = W

Verify by substitution:h(W(h)):\[ h(W(h)) = h \left(50 + 2.3 (h - 60) \right) \]Substitute and simplify:\[ h = \frac{(50 + 2.3 (h - 60)) - 50}{2.3} + 60 = \frac{2.3 (h - 60)}{2.3} + 60 = h - 60 + 60 = h \]W(h(W)):\[ W(h(W)) = W \left( \frac{W - 50}{2.3} + 60 \right) \]Substitute and simplify:\[ W = 50 + 2.3 \left( \frac{W - 50}{2.3} \right) = 50 + 2.3 \left( \frac{W - 50}{2.3} + (60 - 60) \right) = W \]Both equalities are satisfied, verifying that h = h(W) is the inverse of W = W(h).

Determine the height for ideal weight of 80 kg

Using the height function:\[ h(W) = \frac{W - 50}{2.3} + 60 \]Substitute W = 80 into the function:\[ h(80) = \frac{80 - 50}{2.3} + 60 \]Compute the value inside the fraction:\[ 80 - 50 = 30 \]Divide 30 by 2.3:\[ \frac{30}{2.3} \approx 13.04 \]Add 60 to this result:\[ h(80) \approx 13.04 + 60 = 73.04 \text{inches} \]So the height of a male who is at his ideal weight of 80 kilograms is approximately 73.04 inches.

Key concepts

Function Notation

Function notation is an essential concept in mathematics used to describe the relationship between two variables. It helps to clearly represent which variable is dependent and which is independent.
For example, in the function notation \(W(h) = 50 + 2.3(h - 60)\), \(W\) represents the weight (a dependent variable), and \(h\) represents the height (an independent variable). This means that \(W\) depends on the value of \(h\).
Function notation is highly useful because it:

• Provides a clear and concise way to represent the relationship between variables
• Enables easier manipulation and solving of equations
• Makes it intuitive to apply transformations and find values

Understanding function notation is critical for solving algebra problems and understanding how different quantities relate in the real world.

Inverse Notation

Inverse notation allows us to reverse the roles of the dependent and independent variables. This means if we have a function \(W = W(h)\), we can find its inverse \(h = h(W)\).
In inverse notation:

• \(W=W(h)\) represents the initial function
• \(h=h(W)\) represents the inverse function

The inverse function essentially 'undoes' the operation of the original function. In our problem, the inverse function helps us find height based on weight. To find the inverse function:

1. Start with the function: \(W(h) = 50 + 2.3 (h - 60)\)
2. Solve for \(h\) in terms of \(W\): \(h = \frac{W - 50}{2.3} + 60\)

This inverse function is useful for checking the correctness of the original function and for finding unknown values when given the output of the function.

Real-World Applications

Function and inverse function concepts have numerous real-world applications. They help model and solve problems in various fields including:

• Economics: To find costs and revenues based on production units
• Physics: To determine variables like velocity and acceleration
• Biometrics: For calculating body weight or height based on certain data points

In the given problem, we model the ideal body weight of a man based on his height. By reversing this relationship, we can also determine the height of a man from his weight. These applications show the powerful utility of understanding functions and their inverses in solving practical problems.

Problem-Solving in Algebra

Algebra teaches us how to manipulate equations and expressions to find unknown values. Problem-solving in algebra often involves:

• Understanding the function or relationship expressed in a problem
• Applying algebraic techniques to isolate variables
• Verifying solutions by substitution back into the original equations

In the step-by-step solution:

1. We converted height from feet to inches
2. Used the weight function to find the ideal weight
3. Expressed height in terms of weight by manipulating the equation
4. Verified the inverse function relationship
5. Determined height from the ideal weight using the inverse function

Each of these steps demonstrates core algebraic skills that are essential for tackling various types of problems. Algebra provides a systematic approach to problem-solving that is applicable in both academic and real-life contexts.