Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Compute the compositions, \(f(g(x))\), of the following pairs of functions. In each case specify the domain and range of the composite function, and sketch the graph. Your calculator may assist you. For example, the graph of part A can be drawn on the TI- 86 calculator with GRAPH \(, \quad y(x)=, \quad y 1=x^{-2}, \quad y 2=1 /(1+y 1)\) You may wish to suppress the display of \(y 1\) with SELCT in the \(\mathrm{y}(\mathrm{x})=\) menu. a. \(\begin{array}{lllll}f(z) & =\frac{1}{1+z} & \text { b. } & f(z)=\frac{z}{1+z} & \text { c. } & f(z)=5^{z} \\ g(x) & =x^{2} & & g(x)=\frac{x}{1-x} & & g(x)=\log x\end{array}\) \(\begin{array}{llllll}\text { d. } \quad f(z) & =\frac{1}{z} & \text { e. } & f(z)=\frac{z}{1-z} & \text { f. } & f(z)=\log z \\ g(x) & =1+x^{2} & & g(x)=\frac{x}{1+x} & & g(x)=5^{x}\end{array}\) \(\begin{array}{llllll}\text { g. } f(z) & =2^{z} & \text { h. } & f(z)=2^{z} & \text { i. } & f(z)=\log z \\ g(x) & =-x^{2} & & g(x)=-1 / x^{2} & & g(x)=1-x^{2}\end{array}\)

Step-by-Step Solution

Answer

The composite function is \( f(g(x)) = \frac{1}{1+x^2} \) with domain \( x \in \mathbb{R} \) and range \((0, 1]\).

Step 1: Identify the Functions

First, identify the given functions for the problem part you are solving. For example, in part A, we have \( f(z) = \frac{1}{1+z} \) and \( g(x) = x^2 \).

Step 2: Compute the Composite Function

Substitute \( g(x) \) into \( f(z) \). For example, for part A, substitute \( g(x) = x^2 \) into \( f(z) \), yielding:\[ f(g(x)) = \frac{1}{1+x^2} \].

Step 3: Determine the Domain

The domain of \( f(g(x)) \) is determined by the domain of \( g(x) \) and any restrictions imposed by \( f(z) \). For \( f(g(x)) = \frac{1}{1+x^2} \), \( x^2 \) is defined for all real numbers, and since \( x^2 \) is never \(-1\), the domain is \( x \in \mathbb{R} \).

Step 4: Determine the Range

To find the range, consider the values that \( f(g(x)) \) can take. As \( x^2 \) changes from 0 to \( \infty \), \( f(g(x)) = \frac{1}{1+x^2} \) takes values from 1 to 0 (approaching). Thus, the range is \((0, 1]\).

Step 5: Sketch the Graph

Plot the graph of \( f(g(x)) = \frac{1}{1+x^2} \) using a graphing calculator or graphing software. It should appear as a curve approaching the \(x\)-axis and touching at \( y=1 \) when \( x=0 \).

Key Concepts

Domain and Range

When dealing with function composition, understanding the domain and range is crucial. The **domain** is the set of all inputs for which the function is defined. To find the domain of a composite function, we need to consider both functions involved. For example, let's discuss the composition of the functions given:

• Function 1: \(f(z) = \frac{1}{1+z}\)
• Function 2: \(g(x) = x^2\)

The domain of \(g(x) = x^2\) is all real numbers since squares are defined for any real number. However, we must also check the function \(f(z)\), which contains a denominator of \(1+z\). The restriction here would occur if \(z = -1\), but in the composition, this is not possible since \(x^2\) is always non-negative. Consequently, the domain of the composition \(f(g(x)) = \frac{1}{1+x^2}\) is all real numbers, \(x \in \mathbb{R}\).
The **range** is determined by evaluating the outputs that \(f(g(x))\) can achieve. As \(x^2\) varies from 0 to infinity, the values of \(\frac{1}{1+x^2}\) range from 1 down to numbers approaching zero, but never reaching zero. Therefore, the range of the composition \(f(g(x))\) is \((0,1]\).
Being clear about domain and range ensures the composite function is correctly handled and graphically represented.

Graphing Functions

Graphing the composite function does not have to be intimidating. It helps to visualize how the function behaves. Taking our example of \(f(g(x)) = \frac{1}{1+x^2}\), we can break it down step-by-step.

• First, recognize that the graph of \(g(x) = x^2\) gives a parabola opening upwards.
• Input this into \(f(z) = \frac{1}{1+z}\), transforming each value of \(x^2\) to produce \(f(g(x))\).

The result is a symmetric curve that starts at \( (0,1) \) and gently slopes downwards towards the x-axis as \(x\) moves further from zero, showcasing the asymptotic behavior typical of functions that approach a limit but do not touch it, which in this case is the x-axis itself.
Plotting the graph helps confirm the theoretical domain and range. Tools like graphing calculators or software can perform this task and visibly confirm the expected shape, starting from a peak at \(y = 1\) and approaching \(y = 0\) as \(x\) becomes significantly positive or negative.

Step-by-Step Solutions

Understanding the step-by-step solution process can demystify complex function compositions. Let's dissect the process into manageable steps using a practical example from the exercise: **Step 1: Identify the Functions**
List the functions involved. For instance, for part A, recognize \(f(z) = \frac{1}{1+z}\) and \(g(x) = x^2\).
**Step 2: Compute the Composite Function**
Substitute \(g(x)\) into \(f(z)\). Calculate \(f(g(x))\) by replacing \(z\) with \(x^2\). This forms \(f(g(x)) = \frac{1}{1+x^2}\).
**Step 3: Determine the Domain**
Review the domain of both original and composite functions. For \(g(x) = x^2\), \(x\) can be any real number. Considerational restrictions from \(f(z)\) are irrelevant here because \(x^2\) cannot be \(-1\). Thus, the domain remains \(x \in \mathbb{R}\).
**Step 4: Determine the Range**
Analyze the graph's output to deduce the range. Here, as \(x^2\) increases, \(\frac{1}{1+x^2}\) decreases from 1 toward 0, resulting in a range of \((0, 1]\).
**Step 5: Sketch the Graph**
Finally, use tools or graphing methods to visualize \(f(g(x))\). This visualization underscores known concepts and theoretical insights, aiding comprehension. These steps, when followed interactively, provide both clarity and confidence in tackling function composition challenges.