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Chapter 1: Problem 16

Let \(f(x)=2 x+3\) and \(g(x)=x^{3} .\) Find the indicated quantity. a. \((f \circ g)(1)\) b. \((f \circ g)(-2)\)

Short Answer

Expert verified
a. \((f \circ g)(1) = 5\); b. \((f \circ g)(-2) = -13\).

Step by step solution

01

Understand Function Composition

Function composition is the process of applying one function to the result of another function. In notation, \(f \circ g\)(x) is the same as \(f(g(x))\). This means we first apply \(g(x)\) and then use that result to find \(f(x)\).
02

Calculate \(g(1)\) and \(g(-2)\)

Calculate \(g(x)\) for the given values: 1. \(g(1) = 1^3 = 1\). 2. \(g(-2) = (-2)^3 = -8\).
03

Apply \(f\) to the Results from Step 2

Using the results from Step 2, find \(f(g(x))\):1. For \(g(1) = 1\), calculate \(f(1)\) which is \(2(1) + 3 = 5\).2. For \(g(-2) = -8\), calculate \(f(-8)\) which is \(2(-8) + 3 = -16 + 3 = -13\).
04

Write the Final Composed Function Values

The calculated values give us:1. \((f \circ g)(1) = 5\)2. \((f \circ g)(-2) = -13\)

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Key Concepts

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

Understanding Composite Functions
When we talk about **composite functions**, we're discussing a scenario where one function is applied to the results of another. Think of it like a two-step process: first apply one function, and then use that result in another function. For instance, with the notation \((f \circ g)(x)\), this represents \(f(g(x))\).

This operation means that we first take the input \(x\), apply the function \(g\) to it, and then take the result from \(g(x)\) and apply the function \(f\) to that. In our example, \(f(x) = 2x + 3\) and \(g(x) = x^3\), we could then find \((f \circ g)(x)\) by first calculating \(g(x)\) and then using its result in \(f(x)\).
  • Step 1: Determine \(g(x)\) for the given value.
  • Step 2: Use the result from \(g(x)\) to calculate \(f(g(x))\).
Evaluating Functions with Given Values
**Evaluating functions** involves calculating the output of a function for specific input values. In this case, we need to evaluate the inner function \(g\) at certain values, and then use those results to evaluate the composite function. Let's break down the process:

For part **a**, calculate \(g(1)\):
\[ g(1) = 1^3 = 1 \]
This result is then used as input for \(f\):
\[ f(1) = 2(1) + 3 = 5 \]
So, \((f \circ g)(1) = 5\).

For part **b**, calculate \(g(-2)\):
\[ g(-2) = (-2)^3 = -8 \]
Then apply \(f\) to this result:
\[ f(-8) = 2(-8) + 3 = -16 + 3 = -13 \]
Thus, \((f \circ g)(-2) = -13\).
Step-by-Step Math Solutions Made Easy
Breaking problems into **step-by-step solutions** helps in understanding and solving complex math problems, like composite functions. Here's a great way to tackle similar exercises:
  • **Step 1**: Identify and understand each function involved. Know what each individual function does before composing them.
  • **Step 2**: Evaluate the inner function for the given input values. This typically involves plugging numbers into the function and simplifying.
  • **Step 3**: Use this result as the input for the outer function. This is the key part of the composition where you execute the second operation.
  • **Step 4**: Simplify the result to get your final answer.
These clear and straightforward steps help in organizing your thought process and avoid getting lost in more complex problems. Visualizing each step can make the tasks seem less daunting and easier to tackle. Always remember to double-check each part of your calculations to ensure accuracy.

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Most popular questions from this chapter

Donald \(^{57}\) formulated a mathematical model relating the length of bull trout to their weight. He used the equation \(W=4.8 \cdot 10^{-6} L^{3.15},\) where \(L\) is length in millimeters and \(W\) is weight in grams. Plot this graph for \(L \leq 500 .\) What happens to weight when length is doubled?

Assume the linear cost and revenue models applies. An item costs \(\$ 3\) to make. If fixed costs are \(\$ 1000\) and profits are \(\$ 7000\) when 1000 items are made and sold, find the revenue equation.

How much money should a company deposit in an account with a nominal rate of \(8 \%\) compounded quarterly to have \(\$ 100,000\) for a certain piece of machinery in five years?

Boyle and colleagues \(^{55}\) estimated the demand for water clarity in freshwater lakes in Maine. Shown in the figure is the graph of their demand curve. The horizontal axis is given in meters of visibility as provided by the Maine Department of Environmental Protection. (The average clarity of all Maine lakes is \(3.78 \mathrm{~m}\).) The vertical axis is the price decrease of lakeside housing and is given in thousands of dollars per house. Explain what you think is happening when the visibility gets below \(3 \mathrm{~m}\)

Dowdy \(^{52}\) found that the percentage \(y\) of the lesser grain borer beetle initiating flight was approximated by the equation \(y=-240.03+17.83 T-0.29 T^{2}\) where \(T\) is the temperature in degrees Celsius. a. Find the temperature at which the highest percentage of beetles would fly. b. Find the minimum temperature at which this beetle initiates flight. c. Find the maximum temperature at which this beetle initiates flight.
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