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Chapter 3: Problem 30

Find \(f(3), f(0), f(-1),\) and the value of \(x\) for which \(f(x)=-3 x\) $$f(x)=3 x-5$$

Short Answer

Expert verified
The solution is: \(f(3) = 4\), \(f(0) = -5\), \(f(-1) = -8\), and for which \(f(x) = -3x\), the solution is \(x = 5/6\).

Step by step solution

01

Find \(f(3)\)

First, replace \(x\) with 3 in the function: \(f(3) = 3*3 - 5 = 9 - 5 = 4\). This implies that the value of \(f(3)\) equals 4.
02

Find \(f(0)\)

Replace \(x\) with 0 in the function: \(f(0) = 3*0 - 5 = -5\). Thus, the value of \(f(0)\) is -5.
03

Find \(f(-1)\)

Replacing \(x\) with -1 in the equation: \(f(-1) = 3*(-1) - 5 = -3 - 5 = -8\). So, the value of \(f(-1)\) is -8.
04

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

To solve this equation, set \(f(x)\) equals to \(-3x\), i.e., \(3x - 5 = -3x\). This gets us the equation \(3x + 3x = 5\) which simplifies to \(6x = 5\). Dividing both sides of the equation by 6 we are left with \(x = 5/6\).

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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 one of the most fundamental concepts in algebra. They are the simplest type of function and are generally represented by the formula \(f(x)=mx+c\). In this formula:
  • \(m\) is the slope of the function.
  • \(c\) is the y-intercept, where the line crosses the y-axis.
  • \(x\) and \(y\) are variables.
This type of function creates a straight line when graphed. In the given exercise, the linear function is \(f(x) = 3x - 5\). Here, the slope \(m\) is 3, indicating that for each unit increase in \(x\), \(f(x)\) increases by 3 units. The y-intercept \(c\) is -5, showing that the line crosses the y-axis at -5. Understanding linear functions is critical because they can model many real-world situations with constant rates of change.
Function Evaluation
Function evaluation is the process of finding the output of a function for a particular input value. When evaluating a function, you substitute the input value into the function's equation and perform the necessary calculations. Let's look at the steps for evaluating the function from the exercise:
  • To find \(f(3)\), you replace \(x\) with 3 in the function: \(f(3) = 3(3) - 5 = 9 - 5 = 4\).
  • For \(f(0)\), substitute 0 for \(x\): \(f(0) = 3(0) - 5 = -5\).
  • To evaluate \(f(-1)\), input -1 for \(x\): \(f(-1) = 3(-1) - 5 = -3 - 5 = -8\).
Through function evaluation, we can find the specific outputs for any given inputs, which is essential for solving problems and graphing functions.
Solving Equations
Solving equations involves finding the value of the variable that makes the equation true. In the context of the exercise, we were tasked with finding \(x\) such that \(f(x) = -3x\) for the linear function \(f(x) = 3x - 5\). Here's how to tackle it:
  • Set the given function equal to \(-3x\): \(3x - 5 = -3x\).
  • Add \(3x\) to both sides to combine like terms: \(3x + 3x = 5\).
  • This simplifies to \(6x = 5\).
  • Finally, divide both sides by 6 to isolate \(x\): \(x = \frac{5}{6}\).
Solving equations like this one helps us find values of \(x\) for which the function takes on specific values or satisfies certain conditions. It's a vital skill for algebra that drives deeper understanding of function behavior and relationships.

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Most popular questions from this chapter

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. $$A(x, 3) \text { and } B(x-1,-4)$$

Set up a variation equation and solve for the requested value. Assume that the value of a machine varies inversely with its age. If a drill press is worth \(300\)dollar when it is 2 years old, find its value when it is 6 years old. How much has the machine depreciated in those 4 years?

Graph each equation using any method. $$3 y=7$$

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 \(r=10\)

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. \(A(8,-6)\) and the origin
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