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Chapter 3: Problem 34

Find \(f(3), f(0), f(-1),\) and the value of \(x\) for which \(f(x)=-3 x\) $$f(x)=12+3 x$$

Short Answer

Expert verified
The values are \(f(3) = 21\), \(f(0) = 12\), \(f(-1) = 9\), and \(x = -2\) where \(f(x) = -3x\).

Step by step solution

01

Understand the Function

The given function is a linear function \(f(x) = 12+3x\). This is a function of form \(y = mx + b\), where \(m\), the slope, is 3 and \(b\), the y-intercept, is 12.
02

Find Values of \(f(x)\)

Substitute the given \(x\) values into function \(f(x)\):\nFor \(x=3\), \(f(3) = 12+3*3 = 21\), \nFor \(x=0\), \(f(0) = 12+3*0 = 12\), \nFor \(x=-1\), \(f(-1) = 12+3*(-1) = 9
03

Solve for \(x\)

To find the \(x\) value where \(f(x) = -3x\), we set \(f(x)\) equal to \(-3x\) and solve:\n\(12+3x = -3x\), \n6x = -12, thus \(x = -2\)

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Key Concepts

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

Understanding Function Evaluation
Function evaluation is like inputting values into a machine to see what comes out. For the given function, \( f(x) = 12 + 3x \), you simply substitute the given \( x \) values to find their corresponding outputs. By doing this:
  • For \( x = 3 \), substitute 3 in: \( f(3) = 12 + 3 \times 3 = 21 \).
  • For \( x = 0 \), substitute 0 in: \( f(0) = 12 + 3 \times 0 = 12 \).
  • For \( x = -1 \), substitute -1 in: \( f(-1) = 12 + 3 \times (-1) = 9 \).
This shows how you can find specific values of a function.
Function evaluation gives a clear understanding of how the function behaves with different inputs. You can see the relationship between variables clearly, making it a fundamental part of learning mathematics.
Exploring the Slope-Intercept Form
The function \( f(x) = 12 + 3x \) is in what's known as the slope-intercept form. The general structure for this is \( y = mx + b \). Here:
  • \( m \) is the slope, which is 3. This tells you how steep the line is, showing that for every increase of 1 in \( x \), \( f(x) \) increases by 3.
  • \( b \) is the y-intercept, which is 12. This is where the line crosses the y-axis, at the point (0, 12).
The slope-intercept form makes it easy to graph a linear equation. With the slope, you know how to draw the line, while the y-intercept tells you where to start.
Understanding this form helps you quickly grasp how changes in the function are reflected graphically.
Guidelines for Solving Equations
Solving equations involves finding the value of \( x \) that satisfies the equation. In this case, you want \( f(x) = -3x \). Start by setting the given function equal to \(-3x\) and solve step-by-step:
  • Start with the equation: \( 12 + 3x = -3x \).
  • Add \( 3x \) to both sides to bring all terms involving \( x \) to one side: \( 12 + 3x + 3x = 0 \).
  • Simplify this to \( 12 + 6x = 0 \).
  • Subtract 12 from both sides: \( 6x = -12 \).
  • Finally, divide by 6 to solve for \( x \): \( x = -2 \).
Through these steps, you efficiently solve for \( x \), demonstrating a core skill in algebra. Solving equations is about manipulating the equation to isolate the variable, making it crucial for understanding many mathematical and real-world problems.

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Most popular questions from this chapter

Express each joint variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 3. (OBJECTIVE 3) D varies jointly with \(p\) and \(q\). If \(D=16\) when \(p\) and \(q\) are both 8 , find \(D\) when \(p\) and \(q\) are both 12 .

Express each inverse variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 2. (OBJECTIVE 2) \(r\) varies inversely with \(s .\) If \(r=40\) when \(s=10,\) find \(r\) when \(s=15\).

Express each inverse variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 2. (OBJECTIVE 2) \(V\) varies inversely with \(p .\) If \(V=60\) when \(p=12,\) find \(V\) when \(p=9\).

Fill in the blanks. Assume that \(k\) is a constant. The equation \(y=k x z\) represents _____ variation.

Solve each inequality and graph the solution. $$-5 x+4 \geq-6$$
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