Ideal Body Weight One model for the ideal body weight W for...

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? [Note: The ideal body weight $W$ for women (in kilograms) as a function of height $h$ (in inches) is given by $W(h)=45.5+2.3(h-60) .1$

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? [Note: The ideal body weight $W$ for women (in kilograms) as a function of height $h$ (in inches) is given by $W(h)=45.5+2.3(h-60) .1$

Key Concepts

Inverse Functions

An inverse function reverses the roles of inputs and outputs of the original function; it takes a given output back to the input that produced it. To find an inverse function, one typically swaps the variables and solves for the new output variable. In the given problem, finding the inverse is crucial to express the height in terms of weight and to verify the function reversal property.

Function Composition

Function composition is the process of applying one function to the results of another, denoted as (f ? g)(x) = f(g(x)). It is used to verify the relationship between a function and its inverse; if composed in the correct order, the functions should yield the original input. This concept is applied in the problem to confirm that applying the weight function followed by its inverse (or vice versa) returns the original height or weight.

Linear Functions

A linear function is a function that can be expressed in the form f(x) = mx + b, where m and b are constants. This type of function produces a straight line when graphed and represents a constant rate of change. In the context of the problem, the ideal body weight is modeled as a linear function of height, indicating that for every unit increase in height, the weight increases by a constant amount.

Solving Equations

Solving equations involves manipulating an equation to isolate the variable of interest. In this context, once a function is defined, rearranging the equation to solve for one variable in terms of another is essential. This process is used to express height as a function of weight by isolating the height variable in the model.

Transcript

00:01 Hey guys, so we have this function right here, but w is for weight and kilograms height is 8 in inches.

00:08 Okay, so for question a, what is the ideal weight of a 6 foot mill? so instead of just inputting 6 right away for age, don't do that, we have to convert it into inches, right? so because 1 foot is 12 inches, then 6 feet, 6 feet, should be 72 inches, okay? now let's input this to find the weight.

00:36 Okay, like this.

00:39 Substitute 72 for age here.

00:43 And we'll be able to find the answer.

00:48 Okay, and that should be 77 .6 kilograms.

00:54 All right.

00:54 So about question b, we need to rewrite, more like we need to find the inverse function of the original function.

01:05 Okay.

01:05 So we know that it might be confusing here.

01:09 So to avoid confusion, i'm going to write w of h as y and x for h here.

01:21 Okay, so you might be wondering why would she do that? well, because to find the inverse function, right, we first need to flip x with y.

01:30 I believe that might be more comfortable for you guys.

01:34 Okay so and then all we need to do then is to flip x and y as i just said so do it right here okay and then solve for y and then y will be our inverse function okay divide both size by 2 .3 and then finally add both size by 60 and we'll find our inverse function here so the inverse function is actually 8 in terms of w right so then w will be x here just like this okay so we found our inverse function and now we need to verify it okay first we'll do h of w h right so we're not inputting w here we're actually inputting w of h and what is w of h? w of h is 50 .50 plus 2 .3 times h minus 60 okay, just like this...

Please give Ace some feedback