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Chapter 1: Problem 17

Write a. the sum of the two functions. b. the difference of the first function minus the second function. c. the product of the two functions. d. the quotient of the first function divided by the second function. Evaluate each of these constructed functions at 2. $$ f(x)=5 x+4 ; g(x)=2 x^{2}+7 $$

Short Answer

Expert verified
a. 29, b. -1, c. 210, d. \(\frac{14}{15}\) at \( x=2 \).

Step by step solution

01

Find the Sum of the Functions

To find the sum of the functions, add the functions together: \ \( (f + g)(x) = f(x) + g(x) = (5x + 4) + (2x^2 + 7) = 2x^2 + 5x + 11 \) \ Now evaluate at \( x = 2 \): \ \( (f + g)(2) = 2(2)^2 + 5(2) + 11 = 8 + 10 + 11 = 29 \).
02

Find the Difference of the Functions

To find the difference, subtract the second function from the first: \ \( (f - g)(x) = f(x) - g(x) = (5x + 4) - (2x^2 + 7) = -2x^2 + 5x - 3 \) \ Now evaluate at \( x = 2 \): \ \( (f - g)(2) = -2(2)^2 + 5(2) - 3 = -8 + 10 - 3 = -1 \).
03

Find the Product of the Functions

To find the product, multiply the two functions: \ \( (f \cdot g)(x) = f(x) \cdot g(x) = (5x + 4)(2x^2 + 7) \). \ Expanding this, we get: \ \( (f \cdot g)(x) = 5x(2x^2 + 7) + 4(2x^2 + 7) = 10x^3 + 35x + 8x^2 + 28 = 10x^3 + 8x^2 + 35x + 28 \). \ Now evaluate at \( x = 2 \): \ \( (f \cdot g)(2) = 10(2)^3 + 8(2)^2 + 35(2) + 28 = 80 + 32 + 70 + 28 = 210 \).
04

Find the Quotient of the Functions

To find the quotient, divide the first function by the second: \ \( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{5x + 4}{2x^2 + 7} \). \ Now evaluate at \( x = 2 \): \ \( \left( \frac{f}{g} \right)(2) = \frac{5 \times 2 + 4}{2 \times 2^2 + 7} = \frac{10 + 4}{8 + 7} = \frac{14}{15} \).

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Key Concepts

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

Function Arithmetic
Function arithmetic involves performing operations such as addition, subtraction, multiplication, and division on functions. These operations apply the basic rules of arithmetic to pairs of functions, resulting in new functions. Each of these operations has special rules to consider. For example:
  • **Addition**: To add two functions, simply add their respective terms. For instance, summing functions \(f(x)\) and \(g(x)\) gives \((f+g)(x) = f(x) + g(x)\).
  • **Subtraction**: Subtract one function from another by subtracting corresponding terms, resulting in \((f-g)(x) = f(x) - g(x)\).

  • **Multiplication**: This involves multiplying the functions term by term. The result takes the form \((f\cdot g)(x) = f(x) \cdot g(x)\).
  • **Division**: This is found by dividing one function by another, represented as \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). Division is only possible if the denominator is non-zero.
Function arithmetic allows you to construct new functions from existing ones by manipulating their algebraic expressions. By evaluating these transformed functions at specific points, you gain insights into their behavior.
Polynomial Evaluation
Polynomial evaluation involves finding the value of a polynomial function at a given point. This process is crucial for understanding how functions behave under specific conditions. For instance, if you have a polynomial \( p(x) = 2x^2 + 5x + 11 \), evaluating it at \( x = 2 \) means substituting 2 for every instance of \( x \):
  • Square the \( x \) value: \( 2^2 = 4 \).
  • Multiply by corresponding coefficients and sum: \( 2 \times 4 = 8 \), \( 5 \times 2 = 10 \), and constant \( 11 \).
  • Add these results: \( 8 + 10 + 11 = 29 \).
Evaluating polynomials helps in graphing them and finding specific values efficiently. It shows how simple algebraic manipulation can provide insights into the function鈥檚 output at specific input values.
Quotients in Algebra
In algebra, particularly involving functions, quotients refer to the result of dividing one function by another. When you're asked to find the quotient of two functions such as \( \frac{f(x)}{g(x)} \), there are several things you need to keep in mind:
  • The domain: The function \( g(x) \) must not be zero as division by zero is undefined.
  • Simplification: Sometimes you can simplify the resulting expression by canceling out common factors in the numerator and denominator, if any.
  • Evaluation: Once the quotient function is determined, it can be evaluated at specific points, similar to polynomial evaluation. For example, evaluating at \( x = 2 \) gives a clear numeric result of the division.
In practice, understanding quotients in algebra is vital for solving complex problems, where you must consider constraints and simplifications while performing the division operation.

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Most popular questions from this chapter

A fishing club on the Restigouche River in Canada kept detailed records on the numbers of fish caught by its members between 1880 and \(1930 .\) Average catch tends to be cyclic each decade with the average minimum catch of 0.9 salmon per day occurring in years ending with 0 and average maximum catch of 1.7 salmon per day occurring in years ending with 5 (Source: E. R. Dewey and E. F. Dakin, Cycles: The Science of Prediction. New York: Holt, 1947 ) a. Calculate the period and horizontal shift of the salmon catch cycle. Use these values to calculate the parameters \(b\) and \(c\) for a model of the form \(f(x)=a \sin (b x+c)+d\) b. Calculate the amplitude and average value of the salmon catch cycle. c. Use the constants from parts \(a\) and \(b\) to construct a sine model for the average daily salmon catch.

A company receives \(\$ 2.9\) million for each ship it sells and can build the ships for \(\$ 0.2\) million each. a. What is the company's revenue from building and selling a ship? b. Assuming \(\bar{C}(x)\) represents average cost and \(\bar{P}(x)\) represents average profit when \(x\) ships are built and sold, write an expression for the revenue from the building and selling of \(x\) ships.

In desert areas of the western United States, lizards and other reptiles are harvested for sale as pets. Because reptiles hibernate during the winter months, no reptiles are gathered during the months of January, February, November, and December. The number of lizards harvested during the remaining months can be modeled as $$h(m)=15.3 \sin (0.805 m+2.95)+16.7 $$where \(m\) is the month of the year (i.e., \(m=3\) represents$$\text { March), } 3 \leq m \leq 10$$ (Source: Based on information from the Nevada Division of Wildlife) a. Calculate the amplitude and average value of the model. b. Calculate the highest and lowest monthly harvests. Write a sentence interpreting these numbers in context. c. Calculate the period of the model. Is this model useful for \(m\) outside the stated input range? Explain.

Write the inverse for each function. $$ b(t)=0.5+\frac{5}{0.2 t}, t \neq 0 $$

Describe the graph of a logistic function, using the words concave, inflection, and increasing/decreasing.
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