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Chapter 1: Problem 32

Find a linear function \(g\) given \(g\left(-\frac{1}{4}\right)=-6\) and \(g(2)=3 .\) Then find \(g(-3)\)

Short Answer

Expert verified
The linear function is \(g(x) = 4x - 5\). For \(g(-3)\), the value is \(-17\).

Step by step solution

01

- Determine the form of the linear function

A linear function can be written as: \[ g(x) = mx + c \] where \(m\) is the slope and \(c\) is the y-intercept.
02

- Set up the equations from given points

Insert the given points into the equation: \[ g\left(-\frac{1}{4}\right) = -6 \implies m \left(-\frac{1}{4}\right) + c = -6 \] \[ g(2) = 3 \implies 2m + c = 3 \]
03

- Solve for the slope (m)

We have two equations: 1) \( -\frac{1}{4}m + c = -6 \)2) \( 2m + c = 3 \)Subtract the first equation from the second to eliminate \(c\): \[ 2m + c - \left( -\frac{1}{4}m + c \right) = 3 - (-6) \] \[ 2m + \frac{1}{4}m = 9 \] \[ \frac{9}{4}m = 9 \] \[ m = 4 \]
04

- Solve for the y-intercept (c)

Substitute \(m = 4\) back into one of the original equations. Using \(2m + c = 3\): \[ 2(4) + c = 3 \] \[ 8 + c = 3 \] \[ c = 3 - 8 \] \[ c = -5 \]
05

- Write the equation of the linear function

Now that we have \(m = 4\) and \(c = -5\), the linear function is: \[ g(x) = 4x - 5 \]
06

- Find g(-3)

Substitute \(x = -3\) into the function: \[ g(-3) = 4(-3) - 5 \] \[ g(-3) = -12 - 5 \] \[ g(-3) = -17 \]

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Key Concepts

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

slope-intercept form
The slope-intercept form is a way to write the equation of a straight line. This form is commonly written as:
\[ y = mx + b \]
In this equation, m represents the slope of the line, and b is the y-intercept. The y-intercept is where the line crosses the y-axis.
To use the slope-intercept form, follow these steps:
  • Identify the slope (m) of the line. This describes how steep the line is.
  • Determine the y-intercept (b), which is the point where the line crosses the y-axis.
For example, if you know two points on a line, you can calculate the slope by using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \[ (x_1, y_1) \] and \[ (x_2, y_2) \] are the coordinates of the two points. Once you have the slope and y-intercept, you can write the equation in slope-intercept form.
solving linear equations
Solving linear equations involves finding the values of variables that satisfy the equation. In a linear equation of the form \[ ax + b = c \], the goal is to isolate the variable x.
  • First, rearrange the equation to get all terms containing x on one side and the constant terms on the other.
  • Simplify the equation by combining like terms.
  • Isolate x by performing inverse operations.
For example, to solve \[ 2x + 3 = 7 \], you would subtract 3 from both sides to get: \[ 2x = 4 \] Then, divide both sides by 2 to find: \[ x = 2 \] If you have two equations, you can use methods like substitution or elimination to solve for both variables. For example, in system of equations, you can eliminate one variable by adding or subtracting the equations.
function evaluation
Function evaluation involves finding the value of a function for a given input. This is done by substituting the input value into the function.
For a function f(x), if you want to find f(a) where a is a specific value:
  • Identify the function's equation.
  • Replace x with a in the equation.
  • Simplify the expression to find the result.
For example, if you have a function \[ g(x) = 4x - 5 \] and you want to find g(-3):
1. Substitute -3 for x: \[ g(-3) = 4(-3) - 5 \]
2. Simplify the expression:
\[ g(-3) = -12 - 5 = -17 \] This means that when x = -3, \[ g(x) = -17 \]. Function evaluation is useful for determining specific values and understanding how changes in inputs affect outputs.

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Most popular questions from this chapter

A function \(f\) is a linear function if it can be written as \(f(x)=m x+b,\) where \(m\) and \(b\) are constants. If \(m=0,\) the function is a(n) _____ function \(f(x)=b .\) If \(m=1\) and \(b=0,\) the function is the _____.

Solve and graph the solution set. $$-\frac{3}{4} x \geq-\frac{5}{8}+\frac{2}{3} x$$

Find the center and the radius of the circle. Then graph the circle by hand. Check your graph with a graphing calculator: $$x^{2}+(y-3)^{2}=16$$

Price of a World Series Ticket. In \(1946,\) the lowest price of a World Series ticket was \(\$ 1.20 .\) By 2012 , the lowest price of a ticket had increased to \(\$ 110 .\) (Source: AARP.com) Find the average rate of change in the lowest price of a World Series ticket from 1946 to 2012

Miles makes an investment at \(4 \%\) simple interest. At the end of 1 year, the total value of the investment is \(\$ 1560 .\) How much was originally invested?
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