pre calc homework help

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Practice Test

For the following exercises, determine whether each of the following relations is a function.

y = 2 x + 8 y = 2 x + 8

{ ( 2 , 1 ) , ( 3 , 2 ) , ( − 1 , 1 ) , ( 0 , − 2 ) } { ( 2 , 1 ) , ( 3 , 2 ) , ( − 1 , 1 ) , ( 0 , − 2 ) }

For the following exercises, evaluate the function f ( x ) = − 3 x 2 + 2 x f ( x ) = − 3 x 2 + 2 x at the given input.

f ( −2 ) f ( −2 )

f ( a ) f ( a )

Show that the function f ( x ) = − 2 ( x − 1 ) 2 + 3 f ( x ) = − 2 ( x − 1 ) 2 + 3 is not one-to-one.

Write the domain of the function f ( x ) = 3 − x f ( x ) = 3 − x in interval notation.

Given f ( x ) = 2 x 2 − 5 x , f ( x ) = 2 x 2 − 5 x , find f ( a + 1 ) − f ( 1 ) . f ( a + 1 ) − f ( 1 ) .

Graph the function f ( x ) = { x + 1    if − 2 < x < 3     − x     if   x ≥ 3 f ( x ) = { x + 1    if − 2 < x < 3     − x     if   x ≥ 3

Find the average rate of change of the function f ( x ) = 3 − 2 x 2 + x f ( x ) = 3 − 2 x 2 + x by finding f ( b ) − f ( a ) b − a . f ( b ) − f ( a ) b − a .

For the following exercises, use the functions f ( x ) = 3 − 2 x 2 + x  and  g ( x ) = x f ( x ) = 3 − 2 x 2 + x  and  g ( x ) = x to find the composite functions.

( g ∘ f ) ( x ) ( g ∘ f ) ( x )

( g ∘ f ) ( 1 ) ( g ∘ f ) ( 1 )

Express H ( x ) = 5 x 2 − 3 x 3 H ( x ) = 5 x 2 − 3 x 3 as a composition of two functions, f f and g , g , where ( f ∘ g ) ( x ) = H ( x ) . ( f ∘ g ) ( x ) = H ( x ) .

For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.

f ( x ) = x + 6 − 1 f ( x ) = x + 6 − 1

f ( x ) = 1 x + 2 − 1 f ( x ) = 1 x + 2 − 1

For the following exercises, determine whether the functions are even, odd, or neither.

f ( x ) = − 5 x 2 + 9 x 6 f ( x ) = − 5 x 2 + 9 x 6

f ( x ) = − 5 x 3 + 9 x 5 f ( x ) = − 5 x 3 + 9 x 5

f ( x ) = 1 x f ( x ) = 1 x

Graph the absolute value function f ( x ) = − 2 | x − 1 | + 3. f ( x ) = − 2 | x − 1 | + 3.

Solve | 2 x − 3 | = 17. | 2 x − 3 | = 17.

Solve − | 1 3 x − 3 | ≥ 17. − | 1 3 x − 3 | ≥ 17. Express the solution in interval notation.

For the following exercises, find the inverse of the function.

f ( x ) = 3 x − 5 f ( x ) = 3 x − 5

f ( x ) = 4 x + 7 f ( x ) = 4 x + 7

For the following exercises, use the graph of g g shown in Figure 1 .

On what intervals is the function increasing?

On what intervals is the function decreasing?

Approximate the local minimum of the function. Express the answer as an ordered pair.

Approximate the local maximum of the function. Express the answer as an ordered pair.

For the following exercises, use the graph of the piecewise function shown in Figure 2 .

Find f ( 2 ) . f ( 2 ) .

Find f ( −2 ) . f ( −2 ) .

Write an equation for the piecewise function.

For the following exercises, use the values listed in Table 1 .

Find F ( 6 ) . F ( 6 ) .

Solve the equation F ( x ) = 5. F ( x ) = 5.

Is the graph increasing or decreasing on its domain?

Is the function represented by the graph one-to-one?

Find F − 1 ( 15 ) . F − 1 ( 15 ) .

Given f ( x ) = − 2 x + 11 , f ( x ) = − 2 x + 11 , find f − 1 ( x ) . f − 1 ( x ) .

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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: Precalculus
• Publication date: Oct 23, 2014
• Location: Houston, Texas
• Book URL: https://openstax.org/books/precalculus/pages/1-introduction-to-functions
• Section URL: https://openstax.org/books/precalculus/pages/1-practice-test

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Precalculus Help and Problems

Topics in precalculus will serve as a transition between algebra and calculus , containing material covered in advanced algebra and trigonometry courses. Precalculus consists of insights needed to understand calculus.

Still need help after using our precalculus resources? Use our service to find a precalculus tutor .

A brief overview of sets, one of the fundamental principles of mathematics. The topic of sets introduces grouping of objects and number classifications.

Exponential Functions

An extension of exponents in terms of functions, as well as introducing the constant e . Topics include exponential growth and decay, solving with logs, compound and continuously compounded interest, and the exponential function of e .

Logarithmic Functions

A more in depth look at logarithms and logarithmic functions, as well as how they relate to exponents. Topics include the standard and natural log, solving for x and the inverse properities of logarithms.

Radical Functions

An introduction to functions with square roots and radicals and how they relate to conic sections. Topics include functions finding zeros of radical functions, functions with square roots and higher roots, and functions with no solution.

Series and Sequences

Sequences and Series always go hand in hand and they introduce the concept of Mathematical Patterns and how to deal with them. This section deals with the different types of Series and Sequences as well as the methods of finding the next term in a sequence or the sum of a series.

The major types of series and sequences include:

Arithmetic Progression

Geometric Progression

Factorials, Permutations and Combinations

Introduction to the factorial notation. The concept of combinations and permutations is introduced and explained.

Binomial Theorem

Statement of the Binomial Theorem. Pascals’ Triangle and its relation to Binomial Theorem. Expanding polynomials using the Binomial Theorem.

Parametric Equations

An overview of the use of parametric equations, including parametrizing functions and finding a function for a set of parametric equations. Topics include parametrizing lines, segments, circles and ellipses, and peicewise functions.

Polar Coordinates

An introduction to the polar coordinate system. Topics include graphing points, converting from rectangular to polar and polar to rectangular coordinates, converting degrees and radians, and polar equations.

Introduction to the concept of matrices. Different types of matrices discussed. Matrix algebra including addition, subtraction and multiplication. Matrix inverses and determinants.

Subpages include:

Matrix Equality

Matrix Addition and Subtraction

Matrix Multiplication

Special Matrices

Inverse Matrix

Reduction to Row Echelon Form

Number Line

• -x+3\gt 2x+1
• x+y+z=25,\:5x+3y+2z=0,\:y-z=6
• y>2x,\:y<-x-3
• long\:division\:\frac{x^{3}+x^{2}}{x^{2}+x-2}
• partial\:fractions\:\frac{x}{(x+1)(x-4)}
• line\:m=4,\:(-1,\:-6)
• (3+2i)(3-2i)
• y=\frac{x^2+x+1}{x}
• f(x)=2x+3,\:g(x)=-x^2+5,\:(f\:(g(x)))

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Get detailed solutions to your math problems with our precalculus step-by-step calculator . practice your math skills and learn step by step with our math solver. check out all of our online calculators here .,  example,  solved problems,  difficult problems.

Here, we show you a step-by-step solved example of logarithmic differentiation. This solution was automatically generated by our smart calculator:

To derive the function $x^x$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

Apply natural logarithm to both sides of the equality

 Intermediate steps

Apply logarithm properties to both sides of the equality

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

Derive both sides of the equality with respect to $x$

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=

The derivative of the linear function is equal to $1$

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

Multiply the fraction by the term $x$

Any expression multiplied by $1$ is equal to itself

Simplify the fraction

Multiply both sides of the equation by $y$

Substitute $y$ for the original function: $x^x$

The derivative of the function results in

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Precalculus Equations Calculator

Knowledge about pre-calculus equations, how to solve pre-calculus equations.

Ready to tackle those pre-calculus equations head-on? Don't worry, we're here to equip you with the knowledge and strategies to become an equation-solving superhero! Solving pre-calculus equations can vary widely based on the type of equation, but here are some general steps:

• Simplify the equation: Combine like terms and eliminate any fractions if possible by multiplying through by the denominator.
• Isolate the variable: Use algebraic operations to get the variable you're solving for on one side of the equation and everything else on the other side.
• Check for special conditions: Look for any square roots or logarithmic terms that might restrict the domain of possible solutions.
• Verify your solutions: Always plug them back into the original equation to make sure they actually work, as some operations might introduce extraneous solutions.

How do you describe the domain from an equation in pre-calculus?

The domain of an equation in pre-calculus is like the acceptable range of values your variable can take without causing any mathematical mayhem. Imagine the equation as a game – the domain tells you which numbers are allowed to 'play' (be plugged into the equation) and get a valid answer. The domain of an equation is all the possible x-values that will give you a valid y-value:

• Look for division by zero: Set the denominator (if any) not equal to zero.
• Check for square roots or even roots: Set the expression inside the root ≥ 0 since you can't take a real root of a negative number.
• Consider the context: Sometimes the domain is restricted by the scenario being modeled by the equation.

How do you find the frequency of an equation in pre-calculus?

In pre-calculus, you might encounter equations that represent periodic phenomena, like sound waves or vibrating springs. The frequency of such an equation tells you how often a certain pattern repeats itself. Imagine the equation as a musical note – the frequency tells you how many times that note vibrates per second, which determines its pitch. Here's how to find it:

• Identify the period: This is the length of one full cycle of the function.
• Calculate the frequency: Frequency is defined as 1 divided by the period of the function. So if the period is T , then frequency f is \frac{1}{T} .

How to change equations to polar in pre-calculus?

Polar equations are a way of describing shapes using angles and distances from a central point, kind of like a cosmic map. Converting a rectangular equation (with x and y variables) to polar form (with r, the distance from the center, and theta, the angle) can be a fun challenge.

To convert an equation from Cartesian (rectangular) coordinates to polar coordinates:

• Substitute: Replace x and y in your Cartesian equation with these expressions.
• Simplify: Combine terms where possible to form a single equation in terms of r and \theta .

How to find a cubic polynomial equation from a graph in pre-calculus?

If you have the graph of a cubic polynomial:

• Identify points: Find points where the graph intersects the x-axis (roots) and the y-axis (y-intercept).
• Formulate the equation: Use the roots to form factors of (x - root), and adjust for vertical stretching/compression if necessary.
• Determine the leading coefficient: If possible, use additional points to solve for any coefficients in front of these factors.

How to find an equation with multiple intercepts in pre-calculus?

Sometimes, you'll encounter equations with multiple x and y intercepts. Finding such an equation involves considering all the clues these intercepts provide:

• List the intercepts: Identify the x-intercepts and y-intercepts from the graph.
• Build the equation: For each x-intercept a, there's a factor (x - a) in the equation.
• Determine the form: Use the y-intercept as the constant term if the equation is in factored form, or adjust your equation to pass through these points.

Tips & Tricks for solving pre-calculus equations

• Factoring: Factoring is one of the essential techniques to master for pre-calculus. Besides the basic factorization, try to identify and apply techniques such as the difference of squares, sum and difference of cubes, and trinomial factoring. More advanced techniques, such as the Rational Root Theorem, are also tremendously useful in working with polynomial equations. With such skills, one can reduce complex equations to simpler forms and proceed with ease in their solutions and understanding.
• Graphing: The equations can be visualized in graph form to get a feel for the nature of the equation. Features such as zeros (roots), maxima, minima, and points of inflection can easily be read from a graph. Inference from the graph can help confirm analytical solutions and also provide excellent representation on aspects such as continuity or asymptotic behavior. For instance, the nature of a function around a vertical asymptote or the periodicity of a trigonometric function can make the solution sets clearer.
• Patterns: Recognizing patterns, or symmetry, in equations and their graphs can greatly cut down on the work involved in the solution process. For example, symmetry about the y-axis (even functions) or origin (odd functions) will make the task of finding roots significantly easier. Also, noticing arithmetic or geometric sequences in series problems can lead you to general formulas much faster.
• Practice: Regular practice is a must for gaining proficiency in the various forms of equations and mathematical models. Work problems of all types—from linear and quadratic equations to more complex logarithmic and exponential functions. Each type of problem reinforces various aspects of your problem-solving tool set. Also work applied problems so you become able to translate real situations into mathematical equations.

Solving pre-calculus equations often feels like a puzzle. The more you practice, the better you get at spotting how to manipulate and solve these mathematical challenges. It's not just about finding the right answers; it’s about understanding why they’re right!

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Precalculus (Eureka Math/EngageNY)

Welcome learners, unit 1: module 1: complex numbers and transformations, unit 2: module 2: vectors and matrices, unit 3: module 3: rational and exponential functions, unit 4: module 4: trigonometry, unit 5: module 5: probability and statistics.

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