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Chapter 2: Problem 4

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$\frac{1}{x^{2}+x}$$

Short Answer

Expert verified
Let \(g(x) = x^2 + x\) and \(f(u) = \frac{1}{u}\).

Step by step solution

01

Identify the inner function

To find functions \(f\) and \(g\) such that \(\frac{1}{x^2+x} = f(g(x))\), we first need to determine a possible inner function \(g(x)\). A good candidate is the expression \(x^2+x\), since it matches the denominator of the original function. Thus, we choose \(g(x) = x^2 + x\).
02

Determine the outer function

With \(g(x) = x^2 + x\) as our inner function, consider the overall function. The original function's outer layer is the fraction with numerator 1 and the inner function as the denominator. Thus, we define the outer function as \(f(u) = \frac{1}{u}\), where \(u\) is the result of \(g(x)\).
03

Verify the composition

To ensure that \(f(g(x))\) correctly reconstructs the given function, substitute \(g(x)\) into \(f(u)\). We have \(f(g(x)) = f(x^2 + x) = \frac{1}{x^2 + x}\), which matches the original function. This confirms that our choice of \(f\) and \(g\) are correct for the composition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Inner Function
In function composition, the **inner function** plays a crucial role in defining part of the output for the overall composite function. Think of the inner function like the "core" that you evaluate first before applying the outer function.

In our exercise, the inner function is identified as \(g(x) = x^2 + x\). This means whenever we're using this composite function, we first calculate the result of \(g(x)\) with the input \(x\).

By selecting \(x^2 + x\) as the inner function, it sets a foundation for the composite function's logic. Once you transform the value of \(x\) using \(g\), you then use this result as the input, or "inner" value, for the outer function.
  • Start with choosing a component of your target function as the inner function.
  • It generally appears inside of the brackets in compositions like \(f(g(x))\).
  • The inner function often simplifies complicated expressions by isolating parts of the function.
Decoding the Outer Function
The **outer function** in a composite function takes the result of the inner function and processes it further to give the final output. It is applied after the inner function in a sequence of operations.

In our example, the outer function \(f(u) = \frac{1}{u}\) serves to provide the final shape to our composite function. It processes the result \(u = g(x) = x^2 + x\) from the inner function, turning it into the overall function \(\frac{1}{x^2 + x}\).

The outer function usually handles operations that affect the overall output more significantly, like division, exponentiation, or other outer wrapping processes:
  • It is often "wrapping" around the inner function鈥檚 result.
  • In formulas like \(f(g(x))\), it's the \(f\), acting on the result of \(g\).
  • Choose or derive the outer function based on how the result of the inner function needs to be manipulated or adjusted.
Exploring Mathematical Functions
**Mathematical functions** are at the heart of algebra and calculus, defining relationships between variables. These functions, like the components in our problem, determine outputs from given inputs through specific rules.

In the context of the problem, mathematical functions like \(g(x)\) and \(f(u)\) work together through composition to create complex relationships. Composition allows us to combine simpler functions to form intricate operations and solve complicated problems.

Key aspects of mathematical functions include:
  • They map inputs (like \(x\) in \(g(x)\)) to outputs (like \(x^2 + x\)).
  • Functions can be composed, meaning the output of one serves as the input to another, such as our \(f(g(x))\).
  • Through composition, complex behaviors and patterns can be modeled and understood.

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Most popular questions from this chapter

The population of a city \(x\) years from now is predicted to be \(P(x)=\sqrt[4]{x^{2}+1}\) million people for \(1 \leq x \leq 5 .\) Find when the population will be growing at the rate of a quarter of a million people per year. [Hint: On a graphing calculator, enter the given population function in \(y_{1}\), use NDERIV to define \(1 / 2\) to be the derivative of \(y_{1}\), and graph both on the window \([1,5]\) by \([0,3]\). Then TRACE along \(y_{2}\) to find the \(x\) -coordinate (rounded to the nearest tenth of a unit) at which the \(y\) -coordinate is \(0.25\). You may have to \(Z O O M\) IN to find the correct \(x\) -value.]

Increasing global temperatures raise sea levels by thermal expansion and the melting of polar ice. Precise predictions are difficult, but a United Nations study predicts a rise in sea level (above the 2000 level of \(L(x)=0.02 x^{3}-0.07 x^{2}+8 x\) centimeters, where \(x\) is the number of decades since 2000 (so, for example, \(x=2\) means the year 2020). Find \(L(10), L^{\prime}(10)\), and \(L^{\prime \prime}(10)\), and interpret your answers. [Note: Rising sea levels could flood many islands and coastal regions.]

Suppose that the quantity described is represented by a function \(f(t)\) where \(t\) stands for time. Based on the description: a. Is the first derivative positive or negative? b. Is the second derivative positive or negative? The temperature is dropping increasingly rapidly.

a. Show that the definition of the derivative applied to the function \(f(x)=\sqrt[3]{x}\) at \(x=0\) gives \(f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{\sqrt[3]{h}}{h}\) b. Use a calculator to evaluate the difference quotient \(\frac{\sqrt[3]{h}}{h}\) for the following values of \(h: 0.1,0.0001\), and \(0.0000001\). [Hint: Enter the calculation into your calculator with \(\bar{h}\) replaced by \(0.1\), and then change the value of \(h\) by inserting zeros.] c. From your answers to part (b), does the limit exist? Does the derivative of \(f(x)=\sqrt[3]{x}\) at \(x=0\) exist? d. Graph \(f(x)=\sqrt[3]{x}\) on the window \([-1,1]\) by \([-1,1]\). Do you see why the slope at \(x=0\) does not exist?

Use the Generalized Power Rule to find the derivative of each function. $$f(x)=\frac{1}{\sqrt{2 x^{2}-7 x+1}}$$
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