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Chapter 2: Problem 19

Determine whether the point lies on the graph of the function. $$(-2,-3) ; f(t)=\frac{|t-1|}{t+1}$$

Short Answer

Expert verified
Since f(-2) = -3 and the y-coordinate of the point (-2, -3) is also -3, we can conclude that the point (-2, -3) lies on the graph of the function \(f(t) = \frac{|t-1|}{t+1}\).

Step by step solution

01

Identify the x-coordinate of the point

The x-coordinate of the point is -2. So we need to find the value of the function f(t) at t = -2.
02

Find the value of the function at t = -2

Evaluate the function f(t) = \(\frac{|t-1|}{t+1}\) at t = -2: f(-2) = \(\frac{|-2-1|}{-2+1}\)
03

Simplify the expression

Simplify the expression as follows: f(-2) = \(\frac{|-3|}{-1}\) = \(\frac{3}{-1}\) f(-2) = -3
04

Compare the y-coordinate of the point with the value of the function at t = -2

The y-coordinate of the point is -3, and we found that f(-2) = -3 as well. Since the y-coordinate of the point is equal to the value of the function evaluated at the x-coordinate of the point, we can conclude that the point (-2, -3) lies on the graph of the function f(t) = \(\frac{|t-1|}{t+1}\).

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Key Concepts

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

Function Evaluation
Function evaluation is like checking if a certain input leads to a specific output in a given function. Suppose we have a function, such as \(f(t) = \frac{|t-1|}{t+1}\). This function takes an input, \(t\), and processes it through a formula to produce an output.
For the problem given, we want to see what happens when we plug \(t = -2\) into the function. We follow these steps:
  • Substitute \(-2\) into \(f(t)\): \(f(-2)\).
  • Calculate what the function does to \(-2\): \(f(-2) = \frac{|-2-1|}{-2+1}\).
  • Simplify the expression to find the result: \(f(-2) = -3\).
This evaluation tells us that the output when \(t = -2\) is \(-3\). Evaluating functions is crucial for determining how functions behave at different points.
Absolute Value Function
Absolute value is a concept in mathematics that measures the distance of a number from zero on the number line. This measurement is always non-negative, meaning it changes any negative number to positive.

In our example, we have the absolute value inside the function: \(|t-1|.\) Here's what happens during function evaluation:
  • Calculate \(t-1\) for the given \(t = -2\). The result is \(-3\).
  • Take the absolute value of \(-3\), which is \(|-3| = 3\).
The absolute value operation ensures that any negative input becomes positive, which helps to measure the magnitude without considering direction. This concept is often used in calculus and algebra to ensure calculations remain positive even if the original values were negative.
Point on a Graph
A point on a graph is simply a pair of coordinates, written as \((x, y)\), representing a specific location on a two-dimensional plane. To determine whether a point lies on a graph of a particular function, you check if substituting the x-coordinate into the function gives you the same y-coordinate.

In the original exercise, the point is \((-2, -3)\), and the function is \(f(t) = \frac{|t-1|}{t+1}\). We:
  • Plug in \(t = -2\) into the function to get \(f(-2) = -3\).
  • Compare the calculated output with the y-coordinate \(-3\).
Since the calculated value matches the y-coordinate, \((-2, -3)\) indeed lies on the graph of the function. This process helps confirm the consistency and correctness of graph data points when charting mathematical functions.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(b^{2}=4 a c\), then the graph of the quadratic function \(f(x)=\) \(a x^{2}+b x+c(a \neq 0)\) touches the \(x\) -axis at exactly one point.

Find the vertex, the \(x\) -intercepts (if any), and sketch the parabola. \(f(x)=\frac{3}{8} x^{2}-2 x+2\)

The deaths of children less than 1 yr old per 1000 live births is modeled by the function $$ R(t)=162.8 t^{-3.025} \quad(1 \leq t \leq 3) $$ where \(t\) is measured in 50 -yr intervals, with \(t=1\) corresponding to 1900 . a. Find \(R(1), R(2)\), and \(R(3)\) and use your result to sketch the graph of the function \(R\) over the domain \([1,3]\). b. What was the infant mortality rate in \(1900 ?\) In \(1950 ?\) In \(2000 ?\)

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The relationship between Cunningham Realty's quarterly profit, \(P(x)\), and the amount of money \(x\) spent on advertising per quarter is described by the function $$ P(x)=-\frac{1}{8} x^{2}+7 x+30 \quad(0 \leq x \leq 50) $$ where both \(P(x)\) and \(x\) are measured in thousands of dollars. a. Sketch the graph of \(P\). b. Find the amount of money the company should spend on advertising per quarter in order to maximize its quarterly profits.
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