Problem 123 - Algebra and Trigonometry Real Mathematics, Real People

Question

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((f+g)(-4)\)

Step-by-Step Solution

Answer

\((f+g)(-4) = 69\

Step 1: Compute the sum of the two functions

First, find the sum of the functions f and g. This is accomplished by simply adding the two function rules together: \((f+g)(x) = f(x) + g(x) = (14x - 3) + (8x^2)\). So, \((f+g)(x) = 8x^2 + 14x - 3\).

Step 2: Substitute the specified value into the sum of functions

Next, substitute \(x = -4\) into the function \((f+g)(x)\), we get \((f+g)(-4) = 8*(-4)^2 + 14*(-4) - 3\).

Step 3: Evaluate the expression

Evaluate the expression from the previous step: \((f+g)(-4) = 8*(16) + 14*(-4) - 3 = 128 - 56 - 3 = 69\).

Key Concepts

Composite Functions

Composite functions are functions that are created by combining two or more functions in a specific manner. One of the most common ways to combine functions is through function addition, like in the exercise above.
Composite functions operate by taking the output from one function and using it as the input for another. This chaining of functions is crucial in higher mathematics, especially in calculus where composite functions often appear.

• For example, if you have two functions, \( f(x) \) and \( g(x) \), the composite function \( (f \circ g)(x) \) means that you substitute the input value into \( g(x) \) first, and then take the output of \( g(x) \) and input it into \( f(x) \).
• Function addition, such as \( (f+g)(x) = f(x) + g(x) \), is another way to combine functions by simply adding their expressions.

In the exercise, by finding \((f+g)(x)\), we combined the function expressions of \( f(x) \) and \( g(x) \) to create a new polynomial expression to evaluate further.

Polynomial Evaluation

Polynomial evaluation is the process of calculating the value of a polynomial function for a given input. It is an essential skill in algebra and calculus.
Polynomials are expressions with variables raised to whole number powers, and they often have multiple terms.

• In our given exercise, the polynomial obtained from the function addition is \( 8x^2 + 14x - 3 \).
• To evaluate this polynomial at a specific point, substitute the given value of the variable into the polynomial expression and simplify.

For example, to evaluate the polynomial at \( x = -4 \), substitute \( -4 \) into the polynomial:
\( (f+g)(-4) = 8(-4)^2 + 14(-4) - 3 \).
Calculating this step by step,

• first compute \( 8(-4)^2 = 8 \times 16 = 128 \)
• Next, calculate \(14(-4) = -56 \)
• Finally, subtract \(3 \) from the sum of these values, resulting in \( 69 \).

Understanding how to substitute and simplify helps simplify the process of polynomial evaluation.

Function Notation

Function notation is a systematic way to assign a name to the function, usually represented with letters like \( f \), \( g \), or \( h \), followed by a variable in parentheses such as \( f(x) \).
This notation allows us to express complex mathematical ideas efficiently.

• For example, in the exercise, \( f(x) = 14x - 3 \) defines a linear function named \( f \) with input \( x \).
• Similarly, \( g(x) = 8x^2 \) is another function, but it defines a quadratic form.

Function notation also makes it easy to indicate and perform operations on functions.

• The expression \((f+g)(x)\) specifies the addition of the two functions.
• In function notation, operations like addition, subtraction, multiplication, and division can be indicated by symbols such as \(+\), \(-\), \(\cdot\), and \(/\), making it easier to process mathematical operations.

Learning to interpret and work with function notation is fundamental for understanding more complex mathematical concepts like composite functions and polynomial manipulation.