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Chapter 8: Problem 12

Using the given functions \(f, g,\) and \(h\) where $$ f(x)=x+1 \quad g(x)=e^{x} \quad h(x)=x-2 $$ a. Create the function \(k(x)=(f \circ g \circ h)(x)\). b. Describe the transformation from \(x\) to \(k(x)\).

Short Answer

Expert verified
The function \( k(x) = (f \, \circ \, g \, \circ \, h)(x) = e^{x-2} + 1 \).

Step by step solution

01

Evaluate the innermost function, h(x)

First, evaluate the function h(x) which is given by:\[ h(x) = x - 2 \]
02

Apply g(x) to the result of h(x)

Next, apply the function g(x) to the result from Step 1. Since g(x) is given by:\[ g(x) = e^{x} \],appling this will give us:\[ g(h(x)) = g(x - 2) = e^{x-2} \]
03

Apply f(x) to the result of g(h(x))

Finally, apply the function f(x) to the result from Step 2. f(x) is given by:\[ f(x) = x + 1 \], thus we get:\[ f(g(h(x))) = f(e^{x-2}) = e^{x-2}+1 \]

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Key Concepts

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

Function Evaluation
Function evaluation is like a detailed instruction manual where you follow the steps to get to a final result. In mathematical terms, it means substituting a value or expression into a given function and performing the indicated operations to get the output.

Let's take our function \(h(x) = x - 2\). To evaluate it at a point, say \(x=3\), we substitute and calculate: \(h(3) = 3 - 2 = 1\).
It's crucial to get familiar with this process since we'll stack multiple functions later.
Composite Functions
Composite functions are like a series of machines, where each machine takes an input, processes it, and sends it to the next machine in line. The overall outcome is a combination of all these individual machine processes.

In our exercise, we use three functions:
  • \(f(x) = x + 1\)
  • \(g(x) = e^{x}\)
  • \(h(x) = x - 2\)

We first process \(x\) through \(h(x)\), then through \(g(x)\), and finally through \(f(x)\). This step-by-step application makes a composite function. Mathematically, this is written as \((f \circ g \circ h)(x)\). Here's how it works:
  • Step 1: Apply \(h(x)\): \(x\) becomes \(x - 2\).
  • Step 2: Apply \(g(x)\) to the above result: \(e^{x-2}\).
  • Step 3: Apply \(f(x)\) to \(e^{x-2}\): \(e^{x-2} + 1\).
Exponential Functions
Exponential functions involve the constant \(e\) (approximately 2.718), which is a base for natural logarithms, with an \(x\) as the exponent, which gives it an 'explosive' growth property.
In our example, \(g(x) = e^{x}\) is an exponential function. These functions grow extremely rapidly as \(x\) increases.

When we replace \(x\) with an expression like \(x-2\), the exponential function \(e^{x-2}\) adjusts accordingly. This growth can be visualized easily:
  • For \(x=0\): \(e^{0-2} = e^{-2} \approx 0.1353\)
  • For \(x=2\): \(e^{2-2} = e^{0} = 1\)
  • For \(x=4\): \(e^{4-2} = e^{2} \approx 7.3891\)
Notice the rapid increase! This property makes exponential functions very useful in modeling real-world scenarios like population growth and radioactive decay.

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Most popular questions from this chapter

In Exercises \(16-22,\) show that the two functions are inverses of each other. $$ f(x)=2 x-1 \text { and } g(x)=\frac{x+1}{2} $$

In each part, construct a polynomial function with the indicated characteristics. a. Crosses the \(x\) -axis at least three times b. Crosses the \(x\) -axis at \(-1,3,\) and 10 c. Has a \(y\) -intercept of 4 and degree of 3 d. Has a \(y\) -intercept of -4 and degree of 5

a. (Graphing program required.) Use a function graphing program to estimate the \(x\) -intercepts for each of the following. Make a table showing the degree of the polynomial and the number of \(x\) -intercepts. What can you conclude? \(y=2 x+1 \quad y=x^{3}-5 x^{2}+3 x+5\) \(y=x^{2}-3 x-4\) \(y=0.5 x^{4}+x^{3}-6 x^{2}+x+3\) b. Repeat part (a) for the following functions. How do your results compare with those for part (a)? Are there any modifications you need to make to your conclusions in part (a)? \(y=3 x+5\) \(y=x^{3}-2 x^{2}-4 x+8\) \(y=x^{2}+2 x+3\) \(y=(x-2)^{2}(x+1)^{2}\)

(Graphing program required.) If an object is put in an environment with a fixed temperature \(A\) (the "ambient temperature"), then the object's temperature, \(T,\) at time \(t\) is modeled by Newton's Law of Cooling: \(T=A+C e^{-k t},\) where \(k\) is a positive constant. (Note that \(T\) is a function of \(t\) and as \(t \rightarrow+\infty,\) then \(e^{-k t} \rightarrow 0,\) so the temperature \(T\) of the object gets closer and closer to the ambient temperature, A.) A corpse is discovered in a motel room at midnight. The corpse's temperature is \(80^{\circ}\) and the room temperature is \(60^{\circ}\). Two hours later the temperature of the corpse had dropped to \(75^{\circ}\). (Problem adapted from one in the public domain site S.O.S. Math.) a. Using Newton's Law of Cooling, construct an equation to model the temperature \(T\) of the corpse over time, \(t,\) in hours since the corpse was found. b. Then determine the time of death. (Assume the normal body temperature is \(98.6^{\circ} .\) ) c. Graph the function from \(t=-5\) to \(t=5,\) and identify when the person was alive, and the coordinates where the temperature of the corpse was \(98.6^{\circ}, 80^{\circ},\) and \(75^{\circ}\).

Marketing research by a company has shown that the profit, \(P(x)\) (in thousands of dollars), made by the company is related to the amount spent on advertising, \(x\) (in thousands of dollars), by the equation \(P(x)=230+20 x-0.5 x^{2}\). What expenditure (in thousands of dollars) for advertising gives the maximum profit? What is the maximum profit?
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