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Chapter 1: Problem 4

Evaluate \(f(3), f(-1),\) and \(f(0)\). $$f(x)=-4 x-1$$

Short Answer

Expert verified
The values of the function at the given points are: \(f(3)=-13\), \(f(-1)=3\), \(f(0)=-1\).

Step by step solution

01

Evaluate \(f(3)\)

Substitute \(x=3\) into the function:\[f(3)=-4(3)-1=-12-1=-13\]
02

Evaluate \(f(-1)\)

Now substitute \(x=-1\) into the function: \[f(-1)=-4(-1)-1=4-1=3\]
03

Evaluate \(f(0)\)

Finally, substitute \(x=0\) into the function: \[f(0)=-4(0)-1=0-1=-1\]

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Key Concepts

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

Function Evaluation
When we talk about evaluating a function, we are essentially substituting a specific value for the input, usually denoted as "x," into a function's equation. Let's consider the function given in the exercise: - this function is defined as \( f(x) = -4x - 1 \). For example, to evaluate \( f(3) \):
  • "3" is the input value, so we replace every "x" in the function equation with "3".
  • This becomes: \(-4(3) - 1\).
  • Then, calculate the result which leads to \(-13\).
This process is repeated for other values such as \( -1 \) and \( 0 \). A similar substitution and simplification process gives us \( f(-1) = 3 \) and \( f(0) = -1 \). By substituting the values, we "evaluate" the function, helping us determine the output or the y-value for a given input.
Linear Equations
Linear equations form a straight line when graphed on a coordinate plane. They take the form \( ax + b = c \), where "a," "b," and "c" are constants, and "x" is a variable. In the function provided, \( f(x) = -4x - 1 \), the term \(-4x\) shows us that the slope of the line is \(-4\), meaning for every increase of \(1\) in the "x" direction, the value of \(y\) decreases by \(4\). To understand it visually:
  • Plotting different values into the function gives us points like \((3, -13), (-1, 3)\), and \((0, -1)\).
  • These points lie on the line defined by the equation \(y = -4x - 1\).
Grasping linear equations is crucial because they are foundational in algebra, making it easier to handle more complex mathematical models later.
Substitution Method
The substitution method is a technique used primarily in solving systems of equations or evaluating functions by replacing variables with numbers or other expressions. In our context, it involves substituting given values into a function equation to find specific results. Step-by-step, this looks like:
  • Identify which value of "x" you need to evaluate.
  • Replace every instance of "x" in the function with the given number, such as \(3\), \(-1\), or \(0\).
  • Simplify the equation by executing arithmetic operations like multiplication, addition, or subtraction that follow the substitution.
    • For instance, for \(f(x) = -4x - 1\) and \(x = 3\), substitute to get \(-4(3) - 1 = -13\).
This method allows for systematic evaluation, helping solve problems where direct computation may become complex.

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Most popular questions from this chapter

Solve the inequality. Express your answer in interval notation. $$-2 x-1 \geq \frac{x+5}{2}$$

In this set of exercises, you will use absolute value to study real-world problems. A room thermostat is set at \(68^{\circ} \mathrm{F}\) and measures the temperature of the room with an uncertainty of \(\pm 1.5^{\circ} \mathrm{F}\). Assuming the temperature is uniform throughout the room, use absolute value notation to write an inequality for the range of possible temperatures in the room.

Solve the inequality. Express your answer in interval notation. $$x-4<0$$

Graph the function by hand. $$f(x)=\left\\{\begin{array}{ll} -2, & x<-1 \\ |x|, & -1 \leq x \leq 2 \\ 2, & 2

Graph the piecewise-defined function using a graphing utility. The display should be in DOT mode. $$f(x)=\left\\{\begin{array}{ll} x^{2}, & \text { if }-2 \leq x<0 \\ -x+1, & \text { if } 0 \leq x<2.5 \\ x-3.5, & \text { if } x \geq 2.5 \end{array}\right.$$
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