Problem 40 - Calculus for Biology and Medicine

Question

Let $$ f(x)=\frac{2 x}{3+x}, \quad x \geq 0 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) Find the range of \(f(x)\). (c) For which values of \(x\) is \(f(x)=1 ?\) (d) Based on the graph in (a), explain in words why, for any value \(a\) in the range of \(f(x)\), you can find exactly one value \(x \geq 0\) such that \(f(x)=a\). Determine \(x\) by solving \(f(x)=a\).

Step-by-Step Solution

Answer

(a) Graph approaches y=2. (b) Range: [0, 2). (c) x=3. (d) Function is strictly increasing, so x=\frac{3a}{2-a} is unique.

Step 1: Graphing the Function

Plot the function \(f(x) = \frac{2x}{3+x}\) with a graphing calculator by inputting the function and observing its curve in the first quadrant since \(x \geq 0\). The graph should show a curve that begins at the origin \((0, 0)\), rises and approaches a horizontal asymptote at \(y = 2\).

Step 2: Finding the Range

As \(x\) approaches \(0\), \(f(x)\) approaches \(0\), and as \(x\) approaches \(\infty\), \(f(x)\) approaches \(2\). However, \(f(x)\) will never actually reach \(2\), so the range is \(f(x) \in [0, 2)\).

Step 3: Solving for \(x\) When \(f(x)=1\)

Set \(f(x) = 1\) to find \(x\):\[\frac{2x}{3+x} = 1\]Multiply both sides by \(3+x\):\[2x = 3 + x\]Solve for \(x\):\[2x - x = 3\]\[x = 3\]

Step 4: Unique Value for \(f(x)=a\) and Solving \(f(x)=a\) for \(x\)

The function \(f(x) = \frac{2x}{3+x}\) is strictly increasing because its derivative is positive for \(x \geq 0\). Hence, every horizontal line \(y=a\) within the range \(0 \leq a < 2\) intersects the graph of \(f(x)\) exactly once. To solve \(f(x)=a\):\[\frac{2x}{3+x} = a\]Multiply by \(3+x\):\[2x = a(3+x)\]Expand and rearrange:\[2x = 3a + ax\]\[2x - ax = 3a\]\[x(2-a) = 3a\]Divide by \(2-a\):\[x = \frac{3a}{2-a}\]

Step 5: Conclusion

Thus, for any \(a\) in the range \([0, 2)\) of \(f(x)\), there exists a unique \(x\geq0\) given by \(x=\frac{3a}{2-a}\) such that \(f(x)=a\).

Key Concepts

Function Graphing

Graphing a function allows us to visualize mathematical relationships and recognize patterns that can provide deeper insights into a function's behavior. In this instance, using a graphing calculator or software to plot the function \( f(x) = \frac{2x}{3+x} \) with \( x \geq 0 \) helps us see how it behaves as \( x \) changes.
The graph of this function starts at the origin, which is the point \((0,0)\), and has particular characteristics that are crucial for further analysis.
As \( x \) increases, the graph rises but never quite touches the horizontal asymptote at \( y = 2 \). This asymptotic behavior indicates that while the function grows, it will never actually "reach" a \( y \)-value of 2. Observing such details about the graph arms us with valuable foresight about the function's limits and its continuity.

Range of a Function

The range of a function consists of all the possible output values \( y \) that the function can produce. This tells us how the function behaves and helps us determine its limitations. For the function \( f(x) = \frac{2x}{3+x} \), we look at what happens as \( x \) becomes very large or very small (approaching zero, in this case).
As \( x \to 0 \), \( f(x) \) approaches \( 0 \). As \( x \to \infty \), \( f(x) \) moves towards \( 2 \), but it will never actually be equal to \( 2 \) due to the horizontal asymptote.
This analysis reveals that the range of \( f(x) \) is \([0, 2)\). Knowing this is critical because it tells us every value that \( f(x) \) can take, and guides us when solving equations or determining which \( y \) values to use.

Solving Equations

When solving equations, our goal is to find the value of \( x \) that satisfies the equation. For the function \( f(x) = \frac{2x}{3+x} \), suppose we are tasked with finding \( x \) for which \( f(x) = 1 \).
First, set the function equal to 1: \[ \frac{2x}{3+x} = 1 \]
Multiply through by \( 3+x \) to eliminate the fraction:\[ 2x = 3 + x \]
Then, solve for \( x \):\[ 2x - x = 3 \] \[ x = 3 \]
This process of solving equations helps identify specific \( x \)-values for which a certain function equation holds true, broadening our understanding of specific function properties and their results.

Monotonic Functions

A function is described as monotonic if it either never decreases or never increases as \( x \) increases. The function \( f(x) = \frac{2x}{3+x} \) is an example of a strictly increasing function.
This means the function’s derivative is positive, confirming that as \( x \) increases, \( f(x) \) also increases. Because of this property, there will only be one specific \( x \) for each \( f(x) \), making equations easier to solve and interpretations more straightforward.
To show this, if you pick any value \( a \) within the range \([0, 2)\), there's a unique \( x \) such that \( f(x) = a \). Solving \( f(x) = a \) gives:\[ x = \frac{3a}{2-a} \]
This unique solution arises because the function is monotonic, meaning you’ll encounter exactly one \( x \) for every valid output \( a \). This property provides clarity and precision when determining the input for any given output within a function’s range.