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Chapter 12: Problem 67

Evaluate a function. Given \(v(t)=\frac{2 t}{2 t+1},\) find \(v(3)\).

Short Answer

Expert verified
The value of function \(v\) at \(t=3\) is \(\frac{6}{7}\)

Step by step solution

01

Substituting the value of t

Substitute \(t = 3\) into the function \(v(t) = \frac{2t}{2t+1}\) which gives \(v(3) = \frac{2*3}{2*3 + 1}\).
02

Simplifying the fraction

Simplify the fraction: \(v(3) = \frac{6}{6 + 1}\). Further simplify to: \(v(3) = \frac{6}{7}\)

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Key Concepts

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

Substitution
Substitution is a crucial part of evaluating functions, especially when you're trying to find the function's value at a specific point. This involves replacing the variable in the function with a given number or expression. In our example, we are given a function, \( v(t) = \frac{2t}{2t + 1} \), and we want to find \( v(3) \). This means we substitute \( t = 3 \) into the function. By performing the substitution:
  • Replace every instance of \( t \) with 3.
  • The function \( v(t) \) then transforms into \( v(3) = \frac{2(3)}{2(3) + 1} \).
After substitution, the next steps involve simplifying what you've rewritten. This simple algebraic manipulation is a foundational skill in both algebra and function evaluation. It allows you to effectively calculate the function's output for any given input.
Simplifying fractions
Simplifying fractions is the process of reducing a fraction to its simplest form. This involves ensuring that both the numerator and denominator have no common factors other than 1. For the fraction \( \frac{6}{7} \), there isn鈥檛 any more simplifying because 6 and 7 have no common factors besides 1. Let's quickly look at how we got there from our substitution result:
  • From substitution, our function became \( v(3) = \frac{6}{7} \).
  • The numerator 6 and the denominator 7 are already at their simplest since they share no common factors.
Simplifying makes fractions easier to work with and understand. If you face larger numbers, always check for any common factors that can reduce the fraction further. This skill is particularly important in making rational expressions more manageable.
Rational functions
Rational functions are mathematical expressions representing the quotient of two polynomials. In its basic form, such a function appears as \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). In our example, \( v(t) = \frac{2t}{2t + 1} \) is a rational function. When working with rational functions:
  • Note the behavior as the input approaches values causing the denominator to zero, potentially leading to vertical asymptotes or undefined values.
  • Substitution helps evaluate them at specified points, as done for \( v(3) \).
These functions can also be manipulated with operations like addition, subtraction, multiplication, and division, provided you manage the domains where operations are defined. Understanding rational functions is vital, especially in algebra and calculus, as they often appear in various applied contexts.

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Most popular questions from this chapter

Evaluate a function. Given \(P(x)=\frac{4}{2 x+1},\) find \(P(-2)\).

A rescue helicopter is rushing at a constant speed of 150 mph to reach several people stranded in the ocean 11 mi away after their boat sank. The rescuers can determine how far from the victims they are by using the equation \(D=11-2.5 t,\) where \(D\) is the distance in miles and \(t\) is the time elapsed in minutes. Graph this equation for values of \(t\) from 0 to \(4 .\) The point \((3,3.5)\) is on the graph. Write a sentence that describes the meaning of this ordered pair.

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$y=-2 x-1$$

Graph. (GRAPH CANNOT COPY) $$y=-4$$

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$x+3 y=6$$
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