/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 Find \(f(3), f(0), f(-1),\) and ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 36

Find \(f(3), f(0), f(-1),\) and the value of \(x\) for which \(f(x)=-3 x\) $$f(x)=-\frac{1}{3} x+\frac{1}{3}$$

Short Answer

Expert verified
The values are \(f(3)= -\frac{2}{3}\), \(f(0) =\frac{1}{3}\), \(f(-1)=\frac{2}{3}\), and the value of \(x\) for which \(f(x)=-3x\) is \(x = \frac{1}{8}\).

Step by step solution

01

Evaluate \(f(3)\)

Substitute \(x = 3\) into the function: \(f(3)= -\frac{1}{3}*3 +\frac{1}{3} = -1 + \frac{1}{3} = -\frac{2}{3}\)
02

Evaluate \(f(0)\)

Substitute \(x = 0\) into the function: \(f(0)=-\frac{1}{3}*0 +\frac{1}{3} = \frac{1}{3}\)
03

Evaluate \(f(-1)\)

Substitute \(x = -1\) into the function: \(f(-1)=-\frac{1}{3}*(-1)+\frac{1}{3}= \frac{1}{3} +\frac{1}{3}= \frac{2}{3}\)
04

Find \(x\) for \(f(x)=-3x\)

Set \(f(x) = -3x\) and solve for \(x\): \(-\frac{1}{3}x + \frac{1}{3} = -3x \Rightarrow -\frac{1}{3}x +3x = \frac{1}{3}\;\Rightarrow\; (\frac{-1+9}{3})x = \frac{1}{3} \;\Rightarrow\; (\frac{8}{3})x = \frac{1}{3} \;\Rightarrow\; x = \frac{1}{8}\)

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Key Concepts

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

Understanding Linear Functions
Let's begin by exploring the concept of linear functions, which are foundational in mathematics and crucial for understanding various real-world phenomena. A linear function is a function that creates a straight line when graphed. The general form of a linear function is expressed as \(f(x) = mx + b\), where:
  • \(m\) represents the slope, indicating how steep the line is.
  • \(b\) is the y-intercept, showing where the line crosses the y-axis.
In our example, the function is \(f(x) = -\frac{1}{3}x + \frac{1}{3}\). Here, the slope \(m\) is \(-\frac{1}{3}\) meaning the line decreases as you move along the x-axis, and the y-intercept \(b\) is \(\frac{1}{3}\), revealing the initial value when \(x = 0\). Understanding these components helps in predicting how the function behaves or changes with different x-values.
Deciphering Algebraic Expressions
Algebraic expressions are made up of numbers, variables, and operation symbols. They are used to represent various values and relationships mathematically. In the context of our function, the expression \(-\frac{1}{3}x + \frac{1}{3}\) is an algebraic expression:
  • The term \(-\frac{1}{3}x\) includes a variable \(x\) and signifies a variable component that scales with x.
  • The constant term \(\frac{1}{3}\) is a fixed number added to the scaled value of \(x\).
To evaluate the function for specific values, you substitute the given number into the expression in place of \(x\) and perform the operations. This allows us to calculate the function's output for that input.
Solving Equations Involving Functions
Equations involving functions require finding the unknown values that satisfy the mathematical sentences. In our problem, we needed to find \(x\) such that \(f(x) = -3x\). Solving such problems typically involves these steps:
  • Set the function equal to the expression as specified, here \(f(x) = -3x\).
  • Substitute the function expression, resulting in \(-\frac{1}{3}x + \frac{1}{3} = -3x\).
  • Combine like terms and solve for \(x\) by isolating it on one side of the equation.
In this instance, we manipulate the equation to have: \((\frac{-1+9}{3})x = \frac{1}{3}\), simplifying it and solving results in \(x = \frac{1}{8}\). This process showcases how algebraic manipulation and balance can help uncover the solution.

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Most popular questions from this chapter

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(x\) varies directly with \(c^{2}\) and inversely with \(f .\) If \(x=16\) when \(c=8\) and \(f=12,\) find \(x\) when \(c=10\) and \(f=5\).

Express each combined variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 4. (OBJECTIVE 4) \(p\) varies directly with \(q\) and inversely with \(r .\) If \(p=5\) when \(q=1\) and \(r=6,\) find \(p\) when \(q=5\) and \(r=10\)

Express each direct variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 1. (OBJECTIVE 1) \(A\) varies directly with \(z\). If \(A=30\) when \(z=5,\) find \(A\) when \(z=9\).

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. $$P(2,-7) \text { and } Q(-3,12)$$

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(I\) varies jointly with \(r^{2}\) and \(q .\) If \(I=28\) when \(r=2\) and \(q=7,\) find \(I\) when \(r=4\) and \(q=6\).
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