Problem 56 Find the function values. $$ ... [FREE SOLUTION]

91影视

Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 2: Problem 56

Find the function values. $$ f(n)=5 n^{2}+4 n $$ a) $f(0)$ b) $f(-1)$ c) $f(3)$ d) $f(t)$ e) $f(2 a)$ (1) $f(3)-9$

Short Answer

Expert verified

a) $f(0) = 0$, b) $f(-1) = 1$, c) $f(3) = 57$, d) $f(t) = 5t^2 + 4t$, e) $f(2a) = 20a^2 + 8a$, (1) $f(3) - 9 = 48$.

Step by step solution

Evaluate f(0)

Start by substituting 0 for n in the function \[ f(n) = 5n^2 + 4n \]\[ f(0) = 5(0)^2 + 4(0) = 0 \]So, $f(0) = 0$.

Evaluate f(-1)

Substitute -1 for n in the function \[ f(n) = 5n^2 + 4n \]\[ f(-1) = 5(-1)^2 + 4(-1) = 5(1) - 4 = 5 - 4 = 1 \]So, $f(-1) = 1$.

Evaluate f(3)

Substitute 3 for n in the function \[ f(n) = 5n^2 + 4n \]\[ f(3) = 5(3)^2 + 4(3) = 5(9) + 12 = 45 + 12 = 57 \]So, $f(3) = 57$.

Evaluate f(t)

Substitute t for n in the function \[ f(n) = 5n^2 + 4n \]\[ f(t) = 5(t)^2 + 4(t) = 5t^2 + 4t \]So, $f(t) = 5t^2 + 4t$.

Evaluate f(2a)

Substitute 2a for n in the function \[ f(n) = 5n^2 + 4n \]\[ f(2a) = 5(2a)^2 + 4(2a) = 5(4a^2) + 8a = 20a^2 + 8a \]So, $f(2a) = 20a^2 + 8a$.

Evaluate f(3) - 9

We already found that \[ f(3) = 57 \]So, we subtract 9 from this value: \[ f(3) - 9 = 57 - 9 = 48 \]So, $f(3) - 9 = 48$.

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Key Concepts

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

Polynomial Functions

A polynomial function is a type of mathematical function that includes terms with non-negative integer exponents of variables. For example, the function given in the exercise is $ f(n) = 5n^2 + 4n $, where $ 5n^2 $ and $ 4n $ are terms. These types of functions are called 'polynomials' because they contain multiple terms added together.

Important features of polynomial functions include:

Coefficients: The numerical factors in the terms (e.g., 5 and 4 are the coefficients in the exercise).
Exponents: The powers to which the variables are raised (e.g., 2 in $ 5n^2 $).
Variables: The symbols that represent numbers (e.g., n in $ f(n) $).

Understanding polynomial functions is essential since they are ubiquitous in algebra and appear throughout numerous scientific disciplines.

Substitution

Substitution is a fundamental technique in algebra, where you replace a variable with a number or another expression. This method allows you to evaluate functions at specific points.

For example, to find $ f(0) $ from the function $ f(n) = 5n^2 + 4n $, you substitute $ n $ with 0: $ f(0) = 5(0)^2 + 4(0) = 0 $.

Steps for substitution:

Identify the variable to substitute.
Replace the variable in the function with the given number or expression.
Simplify the expression to arrive at the result.

Substitution helps in calculating exact function values and is a stepping stone for more complex algebraic manipulations.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations. It's a way to represent mathematical ideas succinctly.

In our example, the function $ f(n) = 5n^2 + 4n $ is an algebraic expression. It contains:

Variables: n
Constants: Numbers like 5 and 4
Operations: Multiplication and addition

Algebraic expressions are the foundation of algebra and are used to model real-world situations, solve equations, and understand functional relationships. Simplifying and manipulating these expressions is a key skill in mathematics.

Function Notation

Function notation is a way to name a function and its variables in a precise manner. For instance, in the function $ f(n) = 5n^2 + 4n $, $ f $ is the function name, and $ n $ is the variable.

Key points about function notation:

$ f(n) $ reads as 'f of n,' indicating the function f depending on the variable n.
Allows us to express how the function behaves differently with distinct inputs, such as $ f(0), f(-1), $ or $ f(3) $.
Makes it easy to reference and distinguish between multiple functions, like $ f(n) $ and $ g(x) $.

Using function notation is essential in mathematics as it clarifies which variables are being used and helps in systematically solving problems.

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91影视

Find the function values. $$ f(n)=5 n^{2}+4 n $$ a) \(f(0)\) b) \(f(-1)\) c) \(f(3)\) d) \(f(t)\) e) \(f(2 a)\) (1) \(f(3)-9\)

Short Answer

Step by step solution

Evaluate f(0)

Evaluate f(-1)

Evaluate f(3)

Evaluate f(t)

Evaluate f(2a)

Evaluate f(3) - 9

Key Concepts

Polynomial Functions

Substitution

Algebraic Expressions

Function Notation

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