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Chapter 2: Problem 45

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Evaluate \((g \circ f)(-1)\)

Short Answer

Expert verified
The value for \(g \circ f(-1)\) is 0.

Step by step solution

01

Table Lookup for f(-1)

To find \(g \circ f(-1)\), first we need to find the value of \(f(-1)\). From the first table, we can see that \(f(-1) = -2\).
02

Table Lookup for g(f(-1))

Now let's substitute that value into \(g\). We need to find \(g(f(-1)\), or \(g(-2)\). From the second table, we see that \(g(-2) = 0\).
03

Solution for the Composite Function

Therefore, \(g \circ f(-1) = g(-2) = 0\).

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Key Concepts

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

Function Composition
Function composition is a fundamental concept in mathematics where you combine two functions in such a way that the output of the first function becomes the input of the second. In notation, it is typically represented as \((g \circ f)(x)\), which is read as "g of f of x." Here, you'll evaluate the function \(f\) first and then use its output as the input for \(g\).
This process can be thought of as a kind of pipeline or chain reaction:
  • You start with an initial value \(x\).
  • Compute \(f(x)\) to get a result \(y\).
  • Then use this \(y\) to find \(g(y)\).
  • The result is \((g \circ f)(x)\).
This method allows you to create new functions that are derived from existing ones, offering a powerful way to solve complex problems by building on simpler ones.
Table of Values
A table of values is a straightforward way to represent the outcomes of a function. It shows specific input-output pairs, which allows you to look up the value of a function at particular points directly. This can simplify the process of solving function-related problems without having to compute function expressions each time.
For instance, in the exercise, we were given two tables, one for the function \(f\) and another for \(g\). Each row in these tables provides the value of the function for a specific input.
  • For \(f(x)\), you simply find the row where \(x = -1\) and see that \(f(-1) = -2\).
  • Next, use \(-2\) as the input for the \(g(x)\) table, leading to \(g(-2) = 0\).
Tables are especially useful for quickly evaluating compositions like \( (g \circ f)(x) \) because they provide direct access to the needed values.
Function Evaluation
Evaluating functions is about determining the output for a specific input. The function evaluation involves plugging the given input into the function and calculating the output according to the rule or expression that defines the function.
Using the tables provided in the exercise, function evaluation becomes a straightforward task. Here鈥檚 how it works:
  • Look at the variable or the specific input you need to evaluate the function for, such as \(-1\) for \(f(x)\).
  • Access the table to find the corresponding output, like \(f(-1) = -2\) in our case.
  • Use this output to move to the next function, as we did by evaluating \(g\) at \(-2\).
  • Continue this until all compositions or calculations are complete.
Understanding function evaluation helps in working through more complicated compositions and grasping how functions transform inputs into outputs.

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Most popular questions from this chapter

Attendance at Broadway shows in New York can be modeled by the quadratic function \(p(t)=0.0489 t^{2}-0.7815 t+10.31,\) where \(t\) is the number of years since 1981 and \(p(t)\) is the attendance in millions. The model is based on data for the years \(1981-2000 .\) (Source: The League of American Theaters and Producers, Inc.) (a) Use this model to estimate the attendance in the year \(1995 .\) Compare it to the actual value of 9 million. (b) Use this model to predict the attendance for the year 2006 (c) What is the vertex of the parabola associated with the function \(p\), and what does it signify in relation to this problem? (d) Would this model be suitable for predicting the attendance at Broadway shows for the year \(2025 ?\) Why or why not? (e) \(\quad\) Use a graphing utility to graph the function \(p\) What is an appropriate range of values for \(t ?\)

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\frac{1}{x^{2}+1} ; g(x)=\frac{2 x+1}{3 x-1}$$

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=3 x^{2}+2 x$$

The point (2,4) on the graph of \(f(x)=x^{2}\) has been shifted horizontally to the point \((-3,4) .\) Identify the shift and write a new function \(g(x)\) in terms of \(f(x)\).

Suppose that the vertex and an \(x\) -interceptl of the parabola associated with a certain quadratic function are given by (-1,2) and \((4,0),\) respectively. (a) Find the other \(x\) -intercept. (b) Find the equation of the parabola. (c) Check your answer by graphing the function.
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