Problem 41 - Calculus for Biology and Medicine

Question

Let $$ f(x)=\frac{3 x}{1+x}, \quad x \geq 0 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) Find the range of \(f(x)\). (c) For which values of \(x\) is \(f(x)=2\) ? (d) On the basis of 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

The range of \( f(x) \) is \( 0 \leq f(x) < 3 \). For \( f(x) = 2 \), \( x = 2 \). For any \( a \) in the range, there is a unique \( x \) where \( f(x) = a \).

Step 1: Graphing the Function

To graph the function \( f(x) = \frac{3x}{1+x} \), plot it using a graphing calculator or an online graphing tool. This function is defined for \( x \geq 0 \). As \( x \) increases, the graph approaches the horizontal asymptote \( y = 3 \) but never crosses it.

Step 2: Finding the Range of the Function

The range of \( f(x) = \frac{3x}{1+x} \) is determined by the possible values of the function as \( x \) varies over \([0, \infty)\). As \( x \to 0 \), \( f(x) \to 0 \), and as \( x \to \infty \), \( f(x) \to 3 \). Therefore, the range is \( 0 \leq f(x) < 3 \).

Step 3: Solving for f(x) = 2

To find which values of \( x \) satisfy \( f(x) = 2 \), solve the equation \( \frac{3x}{1+x} = 2 \). This simplifies to \( 3x = 2 + 2x \). Solving for \( x \), we find \( x = 2 \).

Step 4: Explaining Uniqueness of Solutions

By analyzing the graph from step 1, the function is increasing and one-to-one for \( x \geq 0 \). This implies that for every value \( a \) in the range \( 0 \leq a < 3 \), there is exactly one corresponding \( x \) such that \( f(x) = a \).

Step 5: Solving f(x) = a for General Solutions

Set \( \frac{3x}{1+x} = a \) and solve for \( x \). This becomes \( 3x = a + ax \). Solving gives \( x = \frac{a}{3-a} \). Since this denominator cannot be zero for \( a < 3 \), \( x \) is valid and unique for each \( a \) in the range.

Key Concepts

Function Graphing

Graphing a rational function like \[ f(x) = \frac{3x}{1+x} \]can reveal a lot about its behavior. Use a graphing calculator or an online graphing tool to visualize this function, focusing on the domain where \( x \geq 0 \).
When you plot \( f(x) \), you'll observe that as \( x \) increases, the graph gets closer to a horizontal line, known as the horizontal asymptote, at \( y = 3 \). However, it does not touch or cross this line.
The function starts at the origin (0,0) and rises toward the asymptote, illustrating its increasing nature. This visual depiction helps us understand how the function behaves as \( x \) grows infinitely large.

Range of a Function

The range of a function describes all the possible values a function can take. For the function \( f(x) = \frac{3x}{1+x} \), understanding the range helps in determining its behavior over its domain.
Here, as \( x \to 0 \), the function \( f(x) \to 0 \). As \( x \to \infty \), \( f(x) \to 3 \). Thus, the function never actually reaches 3, indicating that it asymptotically approaches 3 but doesn't exceed it.
Therefore, the range of this function is:

• From 0 (inclusive) - the lowest point as \( x \) approaches 0.
• To just below 3 - as the function approaches but never reaches 3.

Mathematically, it's written as \( 0 \leq f(x) < 3 \). This range reflects all possible output values of the function.

Solving Equations

Solving equations involving rational functions requires setting the function equal to a given value and isolating the variable. For instance, if we want to find the \( x \) for which \( f(x) = 2 \), we solve the equation \[ \frac{3x}{1+x} = 2 \].
Start by multiplying both sides by \( 1+x \) to eliminate the fraction:\[ 3x = 2 + 2x \]Subtract \( 2x \) from each side to isolate \( x \):\[ 3x - 2x = 2 \]This simplifies to \( x = 2 \). This means at \( x = 2 \), the function \( f(x) \) equals 2. Solving equations in this way is crucial to finding specific values that satisfy the function's value at certain points.

Increasing Function

An increasing function is one where, as \( x \) increases, \( f(x) \) also increases. The function \( f(x) = \frac{3x}{1+x} \) is increasing because for all \( x \geq 0 \), the output value never decreases.
Checking the derivative of the function:\[ f'(x) = \frac{3}{(1+x)^2} \]This derivative is always positive for \( x \geq 0 \), confirming the function's increasing nature.
The fact that it's increasing and one-to-one means for every \( a \) in the range \( 0 \leq a < 3 \), there is precisely one \( x \) such that \( f(x) = a \). This property is very important, especially when trying to find unique solutions to equations within the function's behavior scope.