If $f(x)=\frac{(3+x)}{(x-3)}$, What Is $f(a+2$\]?A. $\frac{3+f(a+2)}{f(a)-1}$ B. $\frac{(5+a)}{(a-1)}$ C. $\frac{(3+a)}{(a-3)}+2$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving them requires a deep understanding of mathematical operations and properties. In this article, we will explore how to solve algebraic expressions, with a focus on the given function f(x)=(3+x)(x−3)f(x)=\frac{(3+x)}{(x-3)}. We will also examine the options provided and determine the correct solution.

Understanding the Function

The given function is f(x)=(3+x)(x−3)f(x)=\frac{(3+x)}{(x-3)}. This function represents a rational expression, where the numerator is a linear expression and the denominator is also a linear expression. To solve for f(a+2)f(a+2), we need to substitute (a+2)(a+2) into the function in place of xx.

Substituting (a+2)(a+2) into the Function

To substitute (a+2)(a+2) into the function, we need to replace xx with (a+2)(a+2). This gives us:

f(a+2)=(3+(a+2))((a+2)−3)f(a+2) = \frac{(3+(a+2))}{((a+2)-3)}

Simplifying the expression, we get:

f(a+2)=(a+5)(a−1)f(a+2) = \frac{(a+5)}{(a-1)}

Evaluating the Options

Now that we have found the value of f(a+2)f(a+2), we can evaluate the options provided.

Option A: 3+f(a+2)f(a)−1\frac{3+f(a+2)}{f(a)-1}

To evaluate this option, we need to substitute the value of f(a+2)f(a+2) into the expression. We get:

3+(a+5)(a−1)(a+5)(a−1)−1\frac{3+\frac{(a+5)}{(a-1)}}{\frac{(a+5)}{(a-1)}-1}

Simplifying the expression, we get:

(3a+8)(a−1)(a+4)(a−1)\frac{\frac{(3a+8)}{(a-1)}}{\frac{(a+4)}{(a-1)}}

Cancelling out the common factor of (a−1)(a-1), we get:

(3a+8)(a+4)\frac{(3a+8)}{(a+4)}

This is not equal to any of the other options, so we can eliminate option A.

Option B: (5+a)(a−1)\frac{(5+a)}{(a-1)}

To evaluate this option, we need to compare it to the value of f(a+2)f(a+2) that we found earlier. We get:

(5+a)(a−1)=(a+5)(a−1)\frac{(5+a)}{(a-1)} = \frac{(a+5)}{(a-1)}

This is equal to the value of f(a+2)f(a+2) that we found earlier, so we can conclude that option B is correct.

Option C: (3+a)(a−3)+2\frac{(3+a)}{(a-3)}+2

To evaluate this option, we need to compare it to the value of f(a+2)f(a+2) that we found earlier. We get:

(3+a)(a−3)+2=(a+5)(a−1)\frac{(3+a)}{(a-3)}+2 = \frac{(a+5)}{(a-1)}

This is not equal to the value of f(a+2)f(a+2) that we found earlier, so we can eliminate option C.

Conclusion

In this article, we explored how to solve algebraic expressions, with a focus on the given function f(x)=(3+x)(x−3)f(x)=\frac{(3+x)}{(x-3)}. We found the value of f(a+2)f(a+2) and evaluated the options provided. We concluded that option B is the correct solution.

Final Answer

The final answer is (5+a)(a−1)\boxed{\frac{(5+a)}{(a-1)}}.

Additional Tips and Tricks

  • When solving algebraic expressions, it's essential to follow the order of operations (PEMDAS).
  • When substituting values into a function, make sure to replace all instances of the variable.
  • When simplifying expressions, look for common factors to cancel out.
  • When evaluating options, compare them to the value of the expression you found earlier.

Common Mistakes to Avoid

  • Not following the order of operations (PEMDAS).
  • Not replacing all instances of the variable when substituting values into a function.
  • Not simplifying expressions by cancelling out common factors.
  • Not comparing options to the value of the expression you found earlier.

Real-World Applications

Algebraic expressions have numerous real-world applications, including:

  • Physics: Algebraic expressions are used to describe the motion of objects and the forces acting upon them.
  • Engineering: Algebraic expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Algebraic expressions are used to model economic systems and make predictions about future trends.

Conclusion

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. PEMDAS stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I substitute values into a function?

A: To substitute values into a function, simply replace the variable (x) with the given value. For example, if we have the function f(x) = 3x + 2 and we want to find f(4), we would substitute 4 in place of x, resulting in f(4) = 3(4) + 2.

Q: How do I simplify expressions?

A: To simplify expressions, look for common factors to cancel out. For example, if we have the expression 6x/2, we can simplify it by cancelling out the common factor of 2, resulting in 3x.

Q: What is the difference between a function and an expression?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). An expression, on the other hand, is a mathematical statement that combines variables, constants, and mathematical operations. While functions can be thought of as a type of expression, not all expressions are functions.

Q: How do I evaluate options in a multiple-choice question?

A: To evaluate options in a multiple-choice question, compare each option to the value of the expression you found earlier. If an option is equal to the value of the expression, it is likely the correct answer.

Q: What are some common mistakes to avoid when solving algebraic expressions?

A: Some common mistakes to avoid when solving algebraic expressions include:

  • Not following the order of operations (PEMDAS)
  • Not replacing all instances of the variable when substituting values into a function
  • Not simplifying expressions by cancelling out common factors
  • Not comparing options to the value of the expression you found earlier

Q: What are some real-world applications of algebraic expressions?

A: Algebraic expressions have numerous real-world applications, including:

  • Physics: Algebraic expressions are used to describe the motion of objects and the forces acting upon them.
  • Engineering: Algebraic expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Algebraic expressions are used to model economic systems and make predictions about future trends.

Q: How can I practice solving algebraic expressions?

A: There are many ways to practice solving algebraic expressions, including:

  • Working through practice problems in a textbook or online resource
  • Using online tools or apps to generate practice problems
  • Joining a study group or working with a tutor to practice solving algebraic expressions
  • Creating your own practice problems to challenge yourself

Q: What are some tips for mastering algebraic expressions?

A: Some tips for mastering algebraic expressions include:

  • Practicing regularly to build your skills and confidence
  • Breaking down complex problems into smaller, more manageable parts
  • Using visual aids, such as graphs or charts, to help you understand the problem
  • Seeking help from a tutor or teacher if you are struggling with a particular concept.