Algebra Class: Functions and Equations

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Questions and Answers

What is the common denominator for the left side of the equation?

$(x^2 - 2)$

$(x - 1)(x + 2)$

$(x^2 + 3x + 2)$

$(x^2 + x - 2)$ (correct)

What can be factored from the expression $x^2 + x - 2$?

$(x - 1)(x - 2)$

$(x + 1)(x - 2)$

$(x - 2)(x + 1)$

$(x - 1)(x + 2)$ (correct)

How can you combine the left side of the equation?

By subtracting the denominators

By expanding all terms

By multiplying both sides by the denominators

By adding the numerators and keeping the common denominator (correct)

What is the result of simplifying the expression $-rac{1}{x + 2} + rac{2}{(x - 1)(x + 2)}$?

$rac{2 - (x + 2)}{(x - 1)(x + 2)}$ Signup and view all the answers

What should be done to isolate variables when working with rational expressions?

Multiply both sides by the least common denominator Signup and view all the answers

What does the equation f(2 - x) + f(x) = 2 imply about the function f(x)?

f(2 - x) equals 2 minus f(x). Signup and view all the answers

From the equation f(-x) + f(x) = 2, what can be inferred about the function's symmetry?

f(x) is an even function. Signup and view all the answers

If f(x) is represented as -x + 3, what is the value of f(1)?

3 Signup and view all the answers

What would be the output of the function f for f(2)?

2 Signup and view all the answers

What is the function property exhibited by the equation $$f(-x) + f(x) = 2$$?

The function is even. Signup and view all the answers

Which of the following satisfies the equation f(2 - x) + f(x) = 2 if f(x) is defined as -x + 3?

f(0) + f(2) = 2 Signup and view all the answers

What operation is performed to simplify $$\frac{4-2-2}{2-2}$$?

It becomes undefined. Signup and view all the answers

In the equation $$3 - x + 0 = -2x + 3$$, what is the value of x when simplified?

x = 2 Signup and view all the answers

What does the expression $$\frac{3-x}{1}$$ simplify to?

3 - x Signup and view all the answers

What is the left-hand side of the equation $$\frac{4-x-1}{2-1} + \frac{4-2-2}{2-2}$$ before substitution?

Undefined Signup and view all the answers

What is the value of $f(2-2)$?

2 Signup and view all the answers

What does the equation $f(-x) + f(x) = 2$ signify about the function $f(x)$?

It is an even function. Signup and view all the answers

What is the solution to the equation $\frac{-2x + 3}{1 - x} = 2$?

1 Signup and view all the answers

What is the first step to solve $\frac{3 + 2^{2-x}}{1 - x^2} = 2$?

Multiply both sides by $1 - x^2$ Signup and view all the answers

What is the output of the function when $x=1$ according to $f(x) = \frac{x - 1}{x + 1}$?

0 Signup and view all the answers

Study Notes

Functional Equations and Their Properties

The equation $f(2-x) = 2$ indicates a symmetry in the function $f$. For any input $2-x$, the output remains constant at 2.
The relationship $f(-x) + f(x) = 2$ suggests that $f$ is symmetric about the line $y=1$, where the sum of values at $x$ and $-x$ equals 2.

Rational Equations

The equation $\frac{-2x + 3}{1 - x} = 2$ can be solved for $x$ by cross-multiplying and simplifying.
Another rational equation, $\frac{3 + 2^{2-x}}{1 - x^2} = 2$, involves exponential and polynomial terms, suggesting the need for careful treatment of domains and values.

Function Definition

The function defined as $f(x) = \frac{x - 1}{x + 1}$ is a rational function that exhibits behavior such as vertical asymptotes and horizontal asymptotes at specific values of $x$.

Combined Rational Functions

The equation $\frac{1}{x^2 + x - 2} + \frac{2}{x^2 + x - 2} = \frac{1}{x - 1} - \frac{1}{x + 2} + \frac{2}{(x - 1)(x + 2)}$ involves simplification by finding a common denominator to combine fractions effectively.

Evaluating Limits and Indeterminate Forms

In the evaluation of $3 - x + 0 = -2x + 3$, one approaches handling limits or existence of values near points causing indeterminacy (e.g., zero in the denominator).
Recognizing $0/0$ as an indeterminate form is crucial in analysis and requires tools like L'Hôpital's rule or algebraic manipulation to resolve.

Simplifying Expressions

The expression $\frac{4 - x - 1}{2-1} + \frac{4-2-2}{2-2} = -2x + 3$ shows algebraic manipulation. The second term approaches an indeterminate condition indicating the need for further insights or limits.
Final simplifications lead to $3 - x = -2x + 3$, highlighting equal conditions for specific values of $x$.

Conclusion on Functional Relationships

Ultimately, both $f(2 - x) + f(x) = 2$ and $f(- x) + f(x) = 2$ illustrate a consistent relationship in the function that maintains resulting properties across transformations of the function's argument.

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Description

Test your understanding of functions and equations with this quiz. Explore various types of equations and their solutions, including functional equations and rational expressions. Perfect for students in Algebra class.