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Chapter 6: Problem 6

For \(f(x, y)=2^{x}-3^{y},\) find \(f(0,2), f(3,1),\) and \(f(2,3)\)

Short Answer

Expert verified
f(0,2) = -8, f(3,1) = 5, f(2,3) = -23

Step by step solution

01

Understand the function

The function given is \( f(x, y) = 2^x - 3^y \). This means for any input values \( x \) and \( y \), you need to compute \( 2^x \) and \( 3^y \) and then subtract \( 3^y \) from \( 2^x \).
02

Calculate \( f(0, 2) \)

Substitute \( x = 0 \) and \( y = 2 \) into the function. Compute: \[ f(0, 2) = 2^0 - 3^2 \] Since \( 2^0 = 1 \) and \( 3^2 = 9 \), we get: \[ f(0, 2) = 1 - 9 = -8 \]
03

Calculate \( f(3, 1) \)

Substitute \( x = 3 \) and \( y = 1 \) into the function. Compute: \[ f(3, 1) = 2^3 - 3^1 \] Since \( 2^3 = 8 \) and \( 3^1 = 3 \), we get: \[ f(3, 1) = 8 - 3 = 5 \]
04

Calculate \( f(2, 3) \)

Substitute \( x = 2 \) and \( y = 3 \) into the function. Compute: \[ f(2, 3) = 2^2 - 3^3 \] Since \( 2^2 = 4 \) and \( 3^3 = 27 \), we get: \[ f(2, 3) = 4 - 27 = -23 \]

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Key Concepts

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

Function Evaluation
Evaluating a function means plugging in specific values for the variables in the given function. For example, in the function \( f(x, y) = 2^x - 3^y \), the inputs \( x \) and \( y \) need to be replaced with specific numbers to find the value of the function at those points.
  • Given the function \( f(x,y) = 2^x - 3^y \),
  • To find \( f(0, 2) \), substitute \( x = 0 \) and \( y = 2 \),
  • Calculate it step-by-step: \( 2^0 \) and \( 3^2 \).

    • Often, the most important part of evaluating a function is making sure each variable is substituted correctly. Double-check your calculations to ensure you aren't making simple errors.
Exponential Function
An exponential function is a mathematical function of the form \( a^x \), where \( a \) is a constant and \( x \) is the exponent. In our example, the function is expressed with exponential terms \( 2^x \) and \( 3^y \).
  • The term \( 2^x \) means 2 raised to the power of \( x \),
  • Similarly, \( 3^y \) means 3 raised to the power of \( y \).

    • It's beneficial to remember some basic exponent rules:
    • Any number raised to the power of 0 equals 1, for example, \( 2^0 = 1 \).
    • Any number raised to the power of 1 equals itself, for example, \( 3^1 = 3 \).
    Knowing these rules can make evaluating exponential functions faster and easier.
Subtraction in Functions
Subtraction in functions means taking the value of one expression and subtracting it from another. Our function \( f(x, y) = 2^x - 3^y \) involves subtracting \( 3^y \) from \( 2^x \).
Here's a step-by-step approach:
  • First, calculate the value of \( 2^x \).
  • Next, calculate the value of \( 3^y \).
  • Finally, subtract the second result (\( 3^y \)) from the first result (\( 2^x \)).

For example:
  • Given \( f(3, 1) \), calculate \( 2^3 = 8 \) and \( 3^1 = 3 \).
  • Then subtract: \( 8 - 3 = 5 \).

Successfully performing subtraction in functions requires careful execution of each step and accurate arithmetic.

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Most popular questions from this chapter

Consider the data in the following table showing the average life expectancy of women in various years. Note that \(x\) represents the actual year. $$\begin{array}{|cc|}\hline & \text { Life Expectancy } \\\\\text { Year, } x & \text { of Women, } y \text { (years) } \\\\\hline 1950 & 71.1 \\\1960 & 73.1 \\\1970 & 74.7 \\\1980 & 77.4 \\\1990 & 78.8 \\\2000 & 79.5 \\\2003 & 80.1 \\\\\hline\end{array}$$ a) Find the regression line, \(y=m x+b.\) b) Use the regression line to predict the life expectancy of women in 2010 and 2015.

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Consider the following data showing the average life expectancy of men in various years. Note that \(x\) represents the actual year. $$\begin{array}{|cc|}\hline & \text { Life Expectancy } \\\\\text { Year, } x & \text { of Men, } y \text { (years) } \\\1950 & 65.6 \\\1960 & 66.6 \\\1970 & 67.1 \\\1980 & 70.0 \\\1990 & 71.8 \\\2000 & 74.1 \\\2003 & 74.8 \\\\\hline\end{array}$$ a) Find the regression line, \(y=m x+b\) b) Use the regression line to predict the life expectancy of men in 2010 and 2015.

The population density of fireflies in a field is given by \(p(x, y)=\frac{1}{100} x^{2} y,\) where \(0 \leq x \leq 30\) and \(0 \leq y \leq 20, x\) and \(y\) are in feet, and \(p\) is the number of fireflies per square foot. Determine the total population of fireflies in this field.
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