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Chapter 4: Problem 7

Find the linear functions satisfying the given conditions. The graph of the inverse function passes through the points (-1,2) and (0,4)

Short Answer

Expert verified
The linear function is \(y = \frac{1}{2}x - 2\).

Step by step solution

01

Understand the relationship between a function and its inverse

First, note that if a point \((a, b)\) lies on the graph of a function, then \((b, a)\) must lie on the graph of its inverse. The inverse function has its x and y coordinates swapped compared to the original function. Therefore, if points \((-1, 2)\) and \((0, 4)\) lie on the inverse function, then \((2, -1)\) and \((4, 0)\) lie on the original function.
02

Identify the linear function parameters

A linear function can be expressed in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. We will use the points \((2, -1)\) and \((4, 0)\) to find \(m\) and \(c\).
03

Calculate the slope \(m\)

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as: \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-1)}{4 - 2} = \frac{1}{2}\]
04

Solve for the y-intercept \(c\)

Substitute the slope \(m = \frac{1}{2}\) and one of the points (e.g., (2, -1)) into the linear equation \(y = mx + c\):\[-1 = \frac{1}{2} \cdot 2 + c \]\[-1 = 1 + c \]\[c = -2 \]
05

Form the equation of the linear function

Now that you have both \(m = \frac{1}{2}\) and \(c = -2\), you can write the equation of the linear function:\[y = \frac{1}{2}x - 2\]

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Key Concepts

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

Linear Functions
Linear functions are fundamental in mathematics and are characterized by their straight-line graphs. These functions can be represented by the equation \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. Linear functions have the following properties:
  • They graph as straight lines.
  • The slope \( m \) indicates the steepness of the line. A positive slope means the line goes upwards, while a negative slope means it goes downwards.
  • The y-intercept \( c \) is the point where the line crosses the y-axis.
Linear functions are incredibly useful because they model relationships where the change is constant between variables. For example, a taxi fare that increases at a fixed rate per mile can be described by a linear function. Recognizing and understanding the characteristics of linear functions makes it easier to grasp more complex mathematical concepts.
Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most straightforward ways to represent a linear function. The general formulation is \( y = mx + c \). Here, \( m \) denotes the slope, showing how much \( y \) changes for each unit increase in \( x \), and \( c \) is the y-intercept, indicating where the line crosses the y-axis.

Let's break down this form:
  • Slope \( (m) \): This value tells us the rate of change of the line. It is calculated as the "rise over run," or the change in \( y \) over the change in \( x \).
  • Y-Intercept \( (c) \): This is the initial value of \( y \) when \( x = 0 \). It provides a starting point on the graph for plotting the line.
The slope-intercept form is widely used for its simplicity and ease of graphing. It allows students to quickly sketch the line on a coordinate plane and understand the linear relationship between \( x \) and \( y \). Knowing how to manipulate and use this form is an essential skill for solving algebraic problems that involve linear equations.
Point-Slope Formula
The point-slope formula is particularly useful when you know a single point on a line and the line's slope. This formula is an alternative way to write the equation of a linear line. It takes the form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.

This format is beneficial in several scenarios:
  • If you need to write the equation of a line quickly when given a point and a slope, this formula is direct and efficient.
  • It's helpful for converting between different forms of a linear equation, such as into the slope-intercept form.
  • The point-slope form also aids in easily visualizing how a line will behave given changes to its slope or through points.
By understanding the point-slope formula, students can deepen their comprehension of linear equations and manipulate them effectively. Applying this knowledge allows for a smoother transition between various representations of linear relationships, an invaluable skill in both academic and real-world contexts.

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Most popular questions from this chapter

The functions \(f, g,\) and h are defined as follows: $$ f(x)=2 x-3 \quad g(x)=x^{2}+4 x+1 \quad h(x)=1-2 x^{2} $$ In each exercise, classify the function as linear, quadratic, or neither. $$g \circ h$$

Let \(f(x)=a x^{2}+b x,\) where \(a \neq 0\) and \(b \neq 0 .\) Find a value for \(b\) such that the equation \(f(f(x))=0\) has exactly three real roots.

The population \(y\) of a colony of bacteria after \(t\) hr is given by \(y=(t+12) /(0.0004 t+0.024) \quad\) where \(t \geq 0\) (a) Find the initial population (that is, the population when \(t=0 \mathrm{hr}\) ). (b) Determine the long-term behavior of the population (as in Example 7 ).

Suppose we have a table of \(x\) -y values with the \(x\) -values equally spaced. This exercise shows that if the first differences of the \(y\) -values are constant, then the table can be generated by a linear function. Note: To show that the data points are generated by a linear function, it is enough to show that the points all lie on one nonvertical line. Consider the following table with the \(x\) -values equally spaced by an amount \(h \neq 0 .\) (We are assuming that \(h\) is nonzero to guarantee that the three \(x\) -values are distinct.) \begin{tabular}{llll} \hline\(x\) & \(a\) & \(a+h\) & \(a+2 h\) \\ \(y\) & \(y_{1}\) & \(y_{2}\) & \(y_{3}\) \\ \hline \end{tabular} lues are con- is a constant. he two data joining the date data three data Assuming that the first differences of the \(y\) -values a stant, we have \(y_{2}-y_{1}=y_{3}-y_{2}=k\), where \(k\) is a con (a) Check that the slope of the line joining the two dat points \(\left(a, y_{1}\right)\) and \(\left(a+h, y_{2}\right)\) is \(k / h\). (b) Likewise, check that the slope of the line joining the two data points \(\left(a+h, y_{2}\right)\) and \(\left(a+2 h, y_{3}\right)\) is \(k / h\). From parts (a) and (b), we conclude that the three data points lie on a nonvertical line, as required. (The slope \(k / h\) is well defined because we assumed \(h \neq 0 .\) )

(a) Is this a quadratic function? Use a graphing utility to draw the graph. (b) How many turning points are there within the given interval? (c) On the given interval, does the function have a maximum value? A minimum value? $$D(x)=\sqrt{x^{2}-x+1}, \quad x \geq 0 \text { (from Example } 3)$$
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