Sunday, August 7, 2022

# Algebra Functions Examples With Answers

## Example : Evaluating Functions For Algebraic Expressions

Simple equations: examples solving a variety of forms | Linear equations | Algebra I | Khan Academy

Find f when f = 5x 7

Write out the function for x using function notation, replacing the x with an empty set of brackets.

Replace the x in the function with the number or algebraic term in the brackets next to the name of the function.

Apply the correct operations to the number or term as appropriate and simplify.

## Summary Of Polynomial Functions

A polynomial function is a function such as a quadratic, cubic, quartic, among others, that only has non-negative integer powers of x. A polynomial of degree n is a function that has the general form:

$$f=a_^n}+a_^}++a_^2}+a_x+a_$$

where the coefficients a are all real numbers. Although the general form looks very complicated, the particular examples are simpler. For example,

$latex f=2^3}+4^2}+2x$

is a polynomial function of degree 3 since 3 is the largest power of the function. And the function:

$$f=-4^5}+2^4}+3^2}+6$$

is a polynomial function of degree 5 since 5 is the greatest power of the function.

## Example: Finding An Equation Of A Function

Express the relationship 2n+6p=12 as a function p=f\left, if possible.

To express the relationship in this form, we need to be able to write the relationship where p is a function of n, which means writing it as p= expression involving n.

\begin& 2n+6p=12\\ & 6p=12 – 2n & & \text2n\text. \\ & p=\frac & & \text. \\ & p=\frac-\frac \\ & p=2-\fracn \end

Therefore, p as a function of n is written as

p=f\left=2-\fracn

#### Analysis of the Solution

It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

Watch this video to see another example of how to express an equation as a function.

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## Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship other times, the table provides a few select examples from a more complete relationship.

Table \ lists the input number of each month , \, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year . Note that, in this table, we define a days-in-a-month function \ where \\) identifies months by an integer rather than by name.

Table \: Months and number of days per month.

 Month number, \

## Evaluating A Function Given In Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppys memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.

The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.

Pet
Beta fish3600

At times, evaluating a function in table form may be more useful than using equations. Here let us call the function P.

The domain of the function is the type of pet and the range is a real number representing the number of hours the pets memory span lasts. We can evaluate the function P at the input value of goldfish. We would write P\left=2160. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function P seems ideally suited to this function, more so than writing it in paragraph or function form.

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## Composition Of Functions Practice Problems

Put into practice what you have learned about the composition of functions by solving the following problems. Choose an answer and check it to see if you got the correct answer. Check out the solved examples above in case you need help.

## Determining Whether A Relation Represents A Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

The domain is \. The range is \.

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter \. Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter \.

A function \ is a relation that assigns a single value in the range to each value in the domain. In other words, no \-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, , is paired with exactly one element in the range, \.

Now lets consider the set of ordered pairs that relates the terms even and odd to the first five natural numbers. It would appear as

Notice that each element in the domain, is not paired with exactly one element in the range, \. For example, the term odd corresponds to three values from the range, \ and the term even corresponds to two values from the range, \. This violates the definition of a function, so this relation is not a function.

Function

Solution

Solution

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## Algebra Functions On Act Math: Lesson And Practice Questions

Functions. Just hearing the word is enough to send some students running for the hills. But never fear! Though function problems are considered some of the more challenging questions on the ACT, this is only due to the fact that most of you will be far more used to dealing with other math topics than you are functions.

On the ACT, question difficulty is categorized by how familiar you are likely to be with any given question, and the only way to combat this challenge is to practice and get used to dealing with questions that are a little less familiar to you. You will generally see 3-4 function questions on any given ACT, so for those of you who are not yet comfortable with functions , this guide is for you.

This will be your complete guide to ACT functions. We’ll walk you through exactly what functions mean, how to use, manipulate, and identify them, and exactly what kind of function problems you’ll see on the ACT.

## Solutions To The Above Questions

• f = 2 + 4 : factor 2 out in the first two terms = 22 – 2) + 4 : add and subtract 2 = 2)2 – 1/2 : complete square and group like terms
• 2x – 2 = x2 – 5x + 4 : substitute y by 2x – 2 x = 1 and x = 6 : solution of quadratic equation and : points of intersection
• -x2 – x – 8 = – : given -x2 – x – 8 = -x2 + 6x – 8 – = 6 : two polynomials are equal if their corresponding coefficients are equal. k = -13 : solve the above for k
• x2 – 2x + y2 + 4y = 11 : Put terms in x together and terms in y together 2 + 2 – 1 – 4 = 11 2 + 2 = 42 : write equation of circle in standard formcenter and radius = 4 : identify center and radius
• 2×2 + 5x – k = 0 : given discriminant = 25 – 4 = 25 + 8k 25 + 8k > 0 : quadratic equations has 2 real solutions when discriminant is positive k > -25/8
• Determinant = -6 – 5k -6 – 5k = 0 : when determinant is equal to zero the system has no solution k = -6/5 : solve for k
• 6×2 – 13x + 5 =
• Note that i4 = 1 Note also that 231 = 4 * 57 + 3 Hence i231 = 57 * i3
• remainder = f = 54 = 1 : remainder theorem
• h = b / 8 = 2 : formula for x coordinate of vertex b = 16 : solve for b y = 4 for x = 2 : the vertex point is a solution to the equation of the parabola 42 – 16 – c = 4 c = -20 : solve for c
• divide P by to obtain x2 – x + 12 P = = : factor the quadratic term the zeros are : 4 , -3 and 2
• 5 < 2x + 2 < 9 : given 3/2 < x < 7/2 the greatest integer value of is 3
• | – x2 + 4x – 4 | : given
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## Function Notation Explanation & Examples

The concept of functions was developed in the seventeenth century when Rene Descartes used the idea to model mathematical relationships in his book Geometry. The term function was then introduced by Gottfried Wilhelm Leibniz fifty years later after publication of Geometry.

Later, Leonhard Euler formalized the usage of functions when he introduced the concept of function notation y = f . It was until 1837 when Peter Dirichlet a German mathematician gave the modern definition of a function.

## What Is A Function

In mathematics, a function is a set of inputs with a single output in each case. Every function has a domain and range. The domain is the set of independent values of the variable x for a relation or a function is defined. In simple words, the domain is a set of x-values that generate the real values of y when substituted in the function.

On the other hand, the range is a set of all possible values that a function can produce. The range of a function can be expressed in interval notation or inform of inequalities.

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## Function Terms And Definitions

Now that we’ve seen what functions do, let’s talk about the pieces of a function.

Functions will be presented to you either by their equations, their tables, or by their graph . Let’s look at a sample function equation and break it down into its components.

An example of a function:

$f = x^2 + 12$

$f$ is the name of the function