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Chapter 1: Problem 6

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\) $$y=\frac{3}{8 x}$$

Short Answer

Expert verified
It is a power function: \(k = \frac{3}{8}\), \(p = -1\).

Step by step solution

01

Understand the form of a power function

A power function is a function of the form \(y = kx^p\), where \(k\) is a constant coefficient, and \(p\) is the power. Our goal is to see if the given function can be written in this form.
02

Rewrite the given function

The given function is \(y = \frac{3}{8x}\). We can rewrite this function by expressing the \(x\) term using a negative exponent. The expression \(\frac{1}{x}\) can be rewritten as \(x^{-1}\). Thus, \(y = \frac{3}{8} \cdot x^{-1}\).
03

Compare with power function form

The rewritten function \(y = \frac{3}{8} x^{-1}\) can be compared to a power function \(y = k x^p\). From this, we identify \(k = \frac{3}{8}\) and \(p = -1\).
04

Final Verification

Check that the function matches the form of a power function. The function \(y = \frac{3}{8} x^{-1}\) is expressed in the form \(y = k x^p\) with \(k = \frac{3}{8}\) and \(p = -1\), confirming that it is indeed a power function.

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Key Concepts

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

Constant Coefficient
In mathematics, a constant coefficient is a fixed number that multiplies a variable raised to a power in a function. When you examine the expression provided in the example,
it is essential to identify any numbers that remain unchanged regardless of the values taken on by the variables.
  • In the power function form, expressed as \(y = k x^p\), the constant coefficient is represented by \(k\).
  • This value does not change as \(x\) varies, setting the amplitude or scale of the power expression.
Considering our specific problem, when rewriting the function \(y = \frac{3}{8} x^{-1}\), the constant coefficient \(k\) is \(\frac{3}{8}\). This tells us how much the function 'scales' the base, \(x\), raised to the power \(-1\). Understanding constant coefficients is crucial for recognizing the features of different functions and how they affect the graph's shape and orientation.
Negative Exponent
Exponents are pivotal in mathematics because they show how many times a number, known as the base, is multiplied by itself. A negative exponent indicates an important variation on this idea:
  • In essence, a negative exponent translates to the inverse of the base raised to the positive equivalent of the power.
  • If you have an expression like \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\).
In the context of our exercise, we encounter \(x^{-1}\), which can be intuitively understood as \(\frac{1}{x}\).
This understanding allows the transformation from \(y = \frac{3}{8}x^{-1}\) to \(y = \frac{3}{8} \cdot \frac{1}{x}\), maintaining consistency with the concept that exponents can indicate division as well as multiplication.
Expression Rewriting
Rewriting expressions is a vital step in algebra to clarify or simplify mathematical problems. It involves transforming the given expression into an alternative form that reveals new information or simplifies the solution process.
  • The key to rewriting effectively is the understanding of mathematical identities and properties, such as those of exponents and fractions.
  • For our problem, rewriting \(y = \frac{3}{8x}\) involved recognizing how the fraction can be expressed using a negative exponent, leading to \(y = \frac{3}{8}x^{-1}\).
This transformation is critical for identifying whether an expression fits a specific type, like a power function.
Rewriting the expression clarifies its structure, enabling identification of the constant coefficient as \(\frac{3}{8}\) and the power as \(-1\).
Thus, rewriting isn't just about changing appearances but a strategic step to solve mathematical queries more efficiently.

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Most popular questions from this chapter

A quantity \(P\) is an exponential function of time \(t .\) Use the given information about the function \(P=P_{b} a^{t}\) to: (a) Find values for the parameters \(a\) and \(P_{0}\). (b) State the initial quantity and the percent rate of growth or decay. \(P_{0} a^{3}=75 \quad\) and \(\quad P_{0} a^{2}=50\)

A sporting goods wholesaler finds that when the price of a product is \(\$ 25,\) the company sells 500 units per week. When the price is \(\$ 30,\) the number sold per week decreases to 460 units. (a) Find the demand, \(q\), as a function of price, \(p\), assuming that the demand curve is linear. (b) Use your answer to part (a) to write revenue as a function of price. (c) Graph the revenue function in part (b). Find the price that maximizes revenue. What is the revenue at this price?

In \(2011,\) the populations of China and India were approximately 1.34 and 1.19 billion people \(^{69},\) respectively. However, due to central control the annual population growth rate of China was \(0.4 \%\) while the population of India was growing by \(1.37 \%\) each year. If these growth rates remain constant, when will the population of India exceed that of China?

In Problems \(25-28,\) put the functions in the form \(P=P_{0} e^{k t}\). $$P=174(0.9)^{t}$$

If the world's population increased exponentially from 5.937 billion in 1998 to 6.771 billion in 2008 and continued to increase at the same percentage rate between 2008 and \(2012,\) calculate what the world's population would have been in \(2012 .\) How does this compare to the Population Reference Bureau estimate of 7.07 billion in July \(2012 ?^{5 x}\).
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