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Chapter 6: Problem 20

A linear function is given. Evaluate the function at the indicated values and then graph the function over its given domain. $$f(x)=-2+1.7 x, x=2,3,4,5,6,7 ; f(2), f(7)$$

Short Answer

Expert verified
\( f(2) = 1.4 \), \( f(7) = 9.9 \); graph is a line with given points.

Step by step solution

01

Understand the Function

The given function is linear and expressed as \( f(x) = -2 + 1.7x \). This means it's a straight line on the graph. You need to evaluate this function at certain values of x: 2, 3, 4, 5, 6, and 7. You'll also specifically find \( f(2) \) and \( f(7) \).
02

Evaluate the Function at x = 2

To find \( f(2) \), substitute 2 for x in the function: \( f(2) = -2 + 1.7 \times 2 \). Calculating this gives \( f(2) = -2 + 3.4 = 1.4 \).
03

Evaluate the Function at x = 3

Substitute 3 for x in the function: \( f(3) = -2 + 1.7 \times 3 \). Calculating this gives \( f(3) = -2 + 5.1 = 3.1 \).
04

Evaluate the Function at x = 4

Substitute 4 for x in the function: \( f(4) = -2 + 1.7 \times 4 \). Calculating this gives \( f(4) = -2 + 6.8 = 4.8 \).
05

Evaluate the Function at x = 5

Substitute 5 for x in the function: \( f(5) = -2 + 1.7 \times 5 \). Calculating this gives \( f(5) = -2 + 8.5 = 6.5 \).
06

Evaluate the Function at x = 6

Substitute 6 for x in the function: \( f(6) = -2 + 1.7 \times 6 \). Calculating this gives \( f(6) = -2 + 10.2 = 8.2 \).
07

Evaluate the Function at x = 7

Substitute 7 for x in the function: \( f(7) = -2 + 1.7 \times 7 \). Calculating this gives \( f(7) = -2 + 11.9 = 9.9 \).
08

Plot the Points and Graph the Function

Now that you have the points (2, 1.4), (3, 3.1), (4, 4.8), (5, 6.5), (6, 8.2), and (7, 9.9), plot these on a graph. The straight line passing through these points is the graph of the linear function \( f(x) = -2 + 1.7x \).
09

Conclusion

The evaluated function values are \( f(2) = 1.4 \) and \( f(7) = 9.9 \). These, along with other evaluated points, help in plotting the graph of the function over the domain \( x = 2 \) to \( x = 7 \).

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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 involves substituting a specific value for the variable, usually denoted as \( x \), in the function equation. In this case, the function is \( f(x) = -2 + 1.7x \), which is a linear function. To evaluate \( f(x) \) at a particular value, substitute the given \( x \) value into the function and perform the necessary arithmetic operations.

Here's how we can evaluate some values:
  • For \( x = 2 \), substitute 2 into the equation: \( f(2) = -2 + 1.7 \times 2 = 1.4 \).
  • For \( x = 3 \), use the function: \( f(3) = -2 + 1.7 \times 3 = 3.1 \).
  • Similarly, evaluate for other values: \( f(4), f(5), f(6), \) and \( f(7) \).
Function evaluation is crucial as it allows us to determine specific points on a graph. This is the first step in graphing any function since it gives us numerical data to plot.
Graphing Functions
Once you have evaluated several points, you can plot them on a graph to visualize the function. Graphing a linear function like \( f(x) = -2 + 1.7x \) is straightforward since it will form a straight line. You only need two points to draw a line, but having more verifies that the calculations are accurate and that the graph truly represents the function.

Here are the typical steps for graphing:
  • Plot each point on the graph, such as (2, 1.4), (3, 3.1), and so on.
  • Draw a line through these plotted points. With linear functions, this line should extend endlessly in both directions.
  • Make sure the line passes through all your calculated points; any deviations might indicate calculation errors.
The straight line indicates that for any change in \( x \), there is a proportional change in \( f(x) \). This linearity demonstrates the core property of linear functions.
Domain of a Function
The domain of a function refers to all the possible input values (\( x \)-values) for which the function is defined. For linear functions such as \( f(x) = -2 + 1.7x \), the domain is typically all real numbers unless specified otherwise. In our exercise, the domain is specifically provided, ranging from \( x = 2 \) to \( x = 7 \).

Understanding the domain is essential because:
  • It helps you know where to evaluate the function.
  • Restricting or defining a domain can sometimes change the appearance of a graph, such as turning a line into a line segment.
  • Ensures that when you graph the function, you only include the relevant part of the graph. This keeps your graph accurate and allows you to focus on the given problem.
By restricting the domain to \( x = 2 \) through \( x = 7 \), the function is evaluated, graphed, and analyzed within these specific limits, reflecting its contextual relevance.

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Most popular questions from this chapter

Consider the following scenario: Even though singapore is generally regarded as being a conservative city-state, it has its share of drug problems. The rate of increase in the number of drug offenders in singapore who used crystal methamphetamine (commonly called "Ice" in singapore) from 2006 to 2016 can be modeled by the rate function $$f(x)=\left\\{\begin{array}{cl} -395+70.1 x, & 6 \leq x \leq 16 \\ 0, & \text { otherwise } \end{array}\right.$$ where \(x\) represents the number of years since 2000 and \(f(x)\) represents the rate of increase in the number of drug offenders who used "Ice" per year. Compute the area under \(f(x)\) on the interval \(6 \leq x \leq 11\), round the value to the nearest whole number, and interpret the result.

A piece wise defined function is given. Evaluate the function at the indicated values and then graph the function over its given domain. $$f(x)=\left\\{\begin{array}{ll} 0.1, & 0 \leq x \leq 10 \\\0, & \text { otherwise }\end{array} ; f(1.1), f(11)\right.$$

Consider the following scenario: The Pick-Chick restaurant charges \(\$ 2.50\) per chicken piece sold on the first 5 pieces and \(\$ 2.00\) per piece thereafter up to 10 pieces. The cost of the chicken is expressed by the piece wise defined function \(f(x)=\left\\{\begin{array}{cl}2.50 x, & x=1,2,3,4,5 \\\ 2.5+2 x, & x=6,7,8,9,10\end{array}\right.\) where \(x\) represents the number of pieces of chicken sold and \(f(x)\) represents cost in dollars. Evaluate \(f(4)\) and interpret the result.

Make a table of function values using the given discrete domain values. Write the values as ordered pairs and then graph the function. $$f(x)=-10+0.6 x^{2}, \quad x=5,6,7,8,9,10$$

Solve the following applications involving the area of a rectangle. A popular smartphone has a screen height of 3.5 inches and a width of 1.9 inches. Calculate the screen's area.
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