Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 7a

Chapter 4, Problem 7a

Solve each problem. During the course of a year, the number of volunteers available to run a food bank each month is modeled by $V (x)$ , where $V (x) = 2 x^{2} - 32 x + 150$ between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, V(x) is modeled by $V (x) = 31 x - 226$ . Find the number of volunteers in each of the following months.
January

Verified step by step guidance

Identify the given piecewise function for the number of volunteers, $V(x)$, where $x$ represents the month number with $x=1$ for January. For January, since $x=1$ falls between January and August, use the first function: $V(x) = 2x^2 - 32x + 150$.

Substitute $x=1$ into the function $V(x) = 2x^2 - 32x + 150$ to find the number of volunteers in January.

Calculate the value inside the function step-by-step: first compute $x^2$, then multiply by 2, next multiply $x$ by 32, and finally perform the addition and subtraction as indicated.

Write the expression after substitution as $V(1) = 2(1)^2 - 32(1) + 150$ and simplify each term accordingly.

Combine all simplified terms to express the number of volunteers in January, which completes the evaluation for $x=1$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. In this problem, V(x) has one formula from January to August and another from August to December. Understanding how to evaluate the correct expression based on the input value x is essential.

Function Evaluation

Function evaluation involves substituting a given input value into the function's formula to find the output. Here, to find the number of volunteers in January (x=1), substitute x=1 into the appropriate formula and simplify to get V(1).

Quadratic and Linear Functions

The function V(x) is quadratic (2x² - 32x + 150) for January to August and linear (31x - 226) for August to December. Recognizing the type of function helps in understanding its behavior and correctly performing calculations.

Recommended video:

05:35

Introduction to Quadratic Equations

Related Practice

Textbook Question

Use synthetic division to perform each division. (x³ + 3x² +11x + 9) / x+1

views

Textbook Question

Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each month is modeled by $V (x),$ where $V (x) = 2 x^{2} - 32 x + 150$ between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, $V (x)$ is modeled by $V (x) = 31 x - 226$ . Find the number of volunteers in each of the following months.

August

views

Textbook Question

Solve each problem. During the course of a year, the number of volunteers available to run a food bank each month is modeled by $V (x)$ , where $V (x) = 2 x^{2} - 32 x + 150$ between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, $V (x)$ is modeled by $V (x) = 31 x - 226$ . Find the number of volunteers in each of the following months.

October

May

Solve each problem. If y varies directly as x, and y=20 when x=4, find y when x = -6.

Textbook Question

Use the graph to solve each equation or inequality. Use interval notation where appropriate. 2(x-2) / {(x-1)(x-3)} = 0