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Chapter 1: Problem 54

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=x^{2}+1 ; g(x)=x-\sqrt{x} $$

Short Answer

Expert verified
\(f(g(x)) = x^2 - 2x\sqrt{x} + x + 1\), \(g(f(x)) = x^2 + 1 - \sqrt{x^2 + 1}\), \(f(f(x)) = x^4 + 2x^2 + 2\).

Step by step solution

01

Understand the Functions

We have two functions: \(f(x) = x^2 + 1\) and \(g(x) = x - \sqrt{x}\). We need to compute three compositions: \(f(g(x))\), \(g(f(x))\), and \(f(f(x))\).
02

Calculate \(f(g(x))\)

We want to find \(f(g(x))\). Substitute \(g(x) = x - \sqrt{x}\) into the function \(f(x)\). So, \(f(g(x)) = f(x - \sqrt{x})\).\\[f(x - \sqrt{x}) = (x - \sqrt{x})^2 + 1 \] \\[= (x^2 - 2x\sqrt{x} + x) + 1 \] \\[= x^2 - 2x\sqrt{x} + x + 1 \] \So, \(f(g(x)) = x^2 - 2x\sqrt{x} + x + 1\).
03

Calculate \(g(f(x))\)

We need to find \(g(f(x))\). Substitute \(f(x) = x^2 + 1\) into \(g(x)\). So, \(g(f(x)) = g(x^2 + 1)\).\\[g(x^2 + 1) = (x^2 + 1) - \sqrt{x^2 + 1} \] \Hence, \(g(f(x)) = x^2 + 1 - \sqrt{x^2 + 1}\).
04

Calculate \(f(f(x))\)

Now, we find \(f(f(x))\) by substituting \(f(x) = x^2 + 1\) into itself.\\[f(f(x)) = f(x^2 + 1) = (x^2 + 1)^2 + 1 \] \\[= (x^4 + 2x^2 + 1) + 1 \] \\[= x^4 + 2x^2 + 2\] \Thus, \(f(f(x)) = x^4 + 2x^2 + 2\).

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Key Concepts

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

Nested Functions
When working with nested functions, you're dealing with the idea of placing one function inside another. It's like putting a series of Russian dolls one inside the other. For example, if we have two functions, let's say \( f(x) = x^2 + 1 \) and \( g(x) = x - \sqrt{x} \), we can create new functions by composition, such as \( f(g(x)) \), \( g(f(x)) \), and \( f(f(x)) \). Each of these compositions involves taking the output of one function and using it as the input for another function.

Here's how it works:
  • For \( f(g(x)) \), you first compute \( g(x) \) and then substitute that entire expression wherever you see \( x \) in \( f(x) \).
  • For \( g(f(x)) \), you first evaluate \( f(x) \) and then input that into \( g(x) \), replacing \( x \) with \( f(x) \).
  • \( f(f(x)) \) requires you to use \( f(x) \) as the argument for \( f(x) \) again, effectively applying the function twice in succession.
This process of nesting helps in understanding how complex functions build on simpler ones.
Polynomial Functions
Polynomial functions are an essential part of algebra and calculus. They are formed from sums of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient, like \( f(x) = x^2 + 1 \). Here, the function is quadratic, meaning its highest exponent is 2.

A polynomial function can be easily manipulated with operations such as addition, subtraction, and multiplication. These operations preserve the polynomial structure, meaning you'll end up with another polynomial. For instance:
  • Adding \( f(x) + f(x) \) simply doubles the coefficients of \( f(x) \).
  • Squaring \( f(x) \) to find \( f(f(x)) \) involves expanding \((x^2 + 1)^2\), resulting in a new polynomial with terms of higher degree.
Polynomials are smooth and continuous, making them a crucial tool for modeling real-world phenomena, like projectile motion or the shape of a roller coaster.
Radical Functions
Radical functions include roots, such as square roots and cubics. These functions involve terms with a radical, like \( g(x) = x - \sqrt{x} \). The presence of a radical term introduces new considerations during calculations.

Here are a few characteristics of radical functions:
  • Radicals can limit the domain of a function, since you typically cannot take the square root of a negative number (in real numbers).
  • Combining radical functions with other types, like polynomial functions in compositions, can lead to interesting results. For example, in \( g(f(x)) = x^2 + 1 - \sqrt{x^2 + 1} \), the polynomial term \( x^2 + 1 \) is still present but is modified by the radical term.
  • Radicals often produce equations that are harder to solve analytically, hence requiring special methods or numerical approximations.
Understanding these properties helps in identifying when radicals may impact calculations, especially in function compositions.

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Most popular questions from this chapter

ENVIRONMENTAL SCIENCE: Wind Energy The use of wind power is growing rapidly after a slow start, especially in Europe, where it is seen as an efficient and renewable source of energy. Global wind power generating capacity for the years 1996 to 2008 is given approximately by \(y=0.9 x^{2}-3.9 x+12.4\) thousand megawatts (MW), where \(x\) is the number of years after 1995 . (One megawatt would supply the electrical needs of approximately 100 homes). a. Graph this curve on the window \([0,20]\) by \([0,300]\). b. Use this curve to predict the global wind power generating capacity in the year \(2015 .\) [Hint: Which \(x\) -value corresponds to \(2015 ?\) Then use TRACE, EVALUATE, or TABLE.] c. Predict the global wind power generating capacity in the year \(2020 .\)

GENERAL: Boiling Point At higher altitudes, water boils at lower temperatures. This is why at high altitudes foods must be boiled for longer times - the lower boiling point imparts less heat to the food. At an altitude of \(h\) thousand feet above sea level, water boils at a temperature of \(B(h)=-1.8 h+212\) degrees Fahrenheit. Find the altitude at which water boils at \(98.6\) degrees Fahrenheit. (Your answer will show that at a high enough altitude, water boils at normal body temperature. This is why airplane cabins must be pressurized - at high enough altitudes one's blood would boil.)

GENERAL: Tsunamis The speed of a tsunami (popularly known as a tidal wave, although it has nothing whatever to do with tides) depends on the depth of the water through which it is traveling. At a depth of \(d\) feet, the speed of a tsunami will be \(s(d)=3.86 \sqrt{d}\) miles per hour. Find the speed of a tsunami in the Pacific basin where the average depth is 15,000 feet.

world population (in millions) since the year 1700 is approximated by the exponential function \(P(x)=522(1.0053)^{x}\), where \(x\) is the number of years since 1700 (for \(0 \leq x \leq 200\) ). Using a calculator, esti mate the world population in the year: 1750

Smoking and Income Based on a recent study, the probability that someone is a smoker decreases with the person's income. If someone's family income is \(x\) thousand dollars, then the probability (expressed as a percentage) that the person smokes is approximately \(y=-0.31 x+40\) (for \(10 \leq x \leq 100)\) a. Graph this line on the window \([0,100]\) by \([0,50]\). b. What is the probability that a person with a family income of $$\$ 40,000$$ is a smoker? [Hint: Since \(x\) is in thousands of dollars, what \(x\) -value corresponds to $$\$ 40,000 ?]$$ c. What is the probability that a person with a family income of $$\$ 70,000$$ is a smoker? Round your answers to the nearest percent.
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