Problem 58
Question
Given that, for all real \(x\), the expression \(\frac{x^{2}-2 x+4}{x^{2}+2 x+4}\) lies between \(\frac{1}{3}\) and 3 . The values between which the expression \(\frac{9 \cdot 3^{2 x}+6 \cdot 3^{x}+4}{9 \cdot 3^{2 x}-6 \cdot 3^{x}+4}\) lies are (A) 0 and 2 (B) \(-1\) and 1 (C) \(-2\) and 0 (D) \(\frac{1}{3}\) and 3 .
Step-by-Step Solution
Verified Answer
(D) \(\frac{1}{3}\) and 3.
1Step 1: Define the given expression
We start with expression 1: \( \frac{x^{2}-2x+4}{x^{2}+2x+4} \) and need to recognize that its values lie between \( \frac{1}{3} \) and 3 for all real \( x \).
2Step 2: Identify the new expression
We have a second expression: \( \frac{9 \cdot 3^{2x}+6 \cdot 3^{x}+4}{9 \cdot 3^{2x}-6 \cdot 3^{x}+4} \). Notice its similarity to the first expression if we set \( x = 3^y \) in the original expression, the numerator and the denominator of the second expression can be transformed to resemble the original expression.
3Step 3: Substitute and transform the expression
By rewriting \( 3^{2x} \) as \( (3^x)^2 \), we let \( x = 3^y \), transforming the new expression into a form similar to the original expression: \( \frac{(3^x)^2 - 2(3^x) + 4}{(3^x)^2 + 2(3^x) + 4} \).
4Step 4: Compare expressions
Notice now that the numerator \( 9 \cdot 3^{2x} + 6 \cdot 3^x + 4 \) corresponds directly with the form \( a^2 + 2a + 4 \) where \( a = 3^x \) and the denominator \( 9 \cdot 3^{2x} - 6 \cdot 3^x + 4 \) corresponds to \( a^2 - 2a + 4 \).
5Step 5: Identify range similarities
Recognize from the problem statement that, since expression \( \frac{y^2-2y+4}{y^2+2y+4} \) over \( x \) lies between \( \frac{1}{3} \) and 3, the similar behavior must occur for the modified, scaled expression with respect to \( 3^x \).
6Step 6: Conclusion on value range
From the above comparisons, we know the range of values for \( \frac{9 \cdot 3^{2x}+6 \cdot 3^{x}+4}{9 \cdot 3^{2x}-6 \cdot 3^{x}+4} \) will similarly lie between \( \frac{1}{3} \) and 3. Thus, answer choice (D) fits.
Key Concepts
Rational ExpressionsTransformation of VariablesInequality Range
Rational Expressions
Rational expressions are a fundamental concept in algebra, with many applications across various fields of mathematics. A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. Understanding these expressions is crucial to solving many algebraic problems.
Rational expressions often appear in the form \( \frac{a(x)}{b(x)} \), where \( a(x) \) and \( b(x) \) are polynomials. The key is to understand how these expressions behave as the variable changes. Since division by zero is undefined, it is important to determine the values of the variable that make the denominator zero, as they are not included in the domain of the expression.
When working with rational expressions, students should get comfortable simplifying them by finding common factors in the numerator and denominator and cancelling them out. This can lead to a simplified expression and facilitate easier manipulation and understanding of the underlying mathematics.
Rational expressions often appear in the form \( \frac{a(x)}{b(x)} \), where \( a(x) \) and \( b(x) \) are polynomials. The key is to understand how these expressions behave as the variable changes. Since division by zero is undefined, it is important to determine the values of the variable that make the denominator zero, as they are not included in the domain of the expression.
When working with rational expressions, students should get comfortable simplifying them by finding common factors in the numerator and denominator and cancelling them out. This can lead to a simplified expression and facilitate easier manipulation and understanding of the underlying mathematics.
Transformation of Variables
In algebra, transformations of variables are used to simplify or alter equations, making them easier to solve. This technique is instrumental when dealing with complex expressions or finding solutions to specific mathematical problems.
Variable transformation involves changing one variable into another to simplify an expression or equation. A classic example is substituting \( x = 3^y \) to transform an expression involving powers of 3. This alteration can turn complex, exponential expressions into simpler polynomial forms. For example, consider transforming the expression \( 9 \cdot 3^{2x} - 6 \cdot 3^x + 4 \). By letting \( a = 3^x \), we can rewrite the expression as \( 9a^2 - 6a + 4 \), a straightforward polynomial in terms of \( a \).
Transformations like these also assist in identifying integral parts of expressions, highlighting areas for simplification or comparison. This method helps align different expressions on common grounds for easier analysis, as demonstrated in the original exercise.
Variable transformation involves changing one variable into another to simplify an expression or equation. A classic example is substituting \( x = 3^y \) to transform an expression involving powers of 3. This alteration can turn complex, exponential expressions into simpler polynomial forms. For example, consider transforming the expression \( 9 \cdot 3^{2x} - 6 \cdot 3^x + 4 \). By letting \( a = 3^x \), we can rewrite the expression as \( 9a^2 - 6a + 4 \), a straightforward polynomial in terms of \( a \).
Transformations like these also assist in identifying integral parts of expressions, highlighting areas for simplification or comparison. This method helps align different expressions on common grounds for easier analysis, as demonstrated in the original exercise.
Inequality Range
Understanding the range of values that an inequality covers is a crucial skill in mathematics, especially when dealing with rational or transformed expressions.
In the context of inequalities, the term "inequality range" refers to the set of values that satisfies a given inequality. For the exercise, this involves ensuring that the expression \( \frac{x^2-2x+4}{x^2+2x+4} \) lies between \( \frac{1}{3} \) and 3. To determine the inequality range, it is vital to identify the limits imposed by the inequality and analyze how these limits translate into the behavior of different expressions derived from or related to the original one.
One way to establish an inequality range is by substituting values or creating a similar expression where these limits are more apparent. This process often includes rewriting the expression or evaluating specific points to ensure all potential values remain within the specified range. By fully grasping the inequality range concept, students can tackle similar algebraic problems with confidence and ease.
In the context of inequalities, the term "inequality range" refers to the set of values that satisfies a given inequality. For the exercise, this involves ensuring that the expression \( \frac{x^2-2x+4}{x^2+2x+4} \) lies between \( \frac{1}{3} \) and 3. To determine the inequality range, it is vital to identify the limits imposed by the inequality and analyze how these limits translate into the behavior of different expressions derived from or related to the original one.
One way to establish an inequality range is by substituting values or creating a similar expression where these limits are more apparent. This process often includes rewriting the expression or evaluating specific points to ensure all potential values remain within the specified range. By fully grasping the inequality range concept, students can tackle similar algebraic problems with confidence and ease.
Other exercises in this chapter
Problem 56
If the roots of the equation \(x^{2}-2 a x+a^{2}+a-3=0\) are real and less than 3 , then (A) \(a4\)
View solution Problem 57
For all real \(x\), the minimum value of \(\frac{1-x+x^{2}}{1+x+x^{2}}\) is (A) 0 (B) \(\frac{1}{3}\) (C) 1 (D) 3
View solution Problem 59
The value of \(k\) for which the number 3 lies between the roots of the equation \(x^{2}+(1-2 k) x+\left(k^{2}-\right.\) \(k-2)=0\) is given by (A) \(25\)
View solution Problem 60
The number of negative integral solutions of \(x^{2} \cdot 2^{x+1}\) \(+2^{|x-3|+2}=x^{2} \cdot 2^{(x-3 \mid+4)}+2^{x-1}\) is (A) 4 (B) 2 (C) 1 (D) 0
View solution