Select the correct relation between E and F. $E = \frac{x}{1+x}$ and $F = \frac{-x}{1-x}$ $x > 1$
Mathematical Comparison of E and F for x > 1
We are given the expressions for E and F:
- $E = \frac{x}{1+x}$
- $F = \frac{-x}{1-x}$
And the condition $x > 1$.
Simplify Expression for F
First, simplify the expression for F:
$F = \frac{-x}{1-x} = \frac{-x \times (-1)}{(1-x) \times (-1)} = \frac{x}{x-1}$
Compare E and F
Now, we need to compare $E = \frac{x}{1+x}$ and $F = \frac{x}{x-1}$ under the condition $x > 1$.
Since $x > 1$, the numerator $x$ is positive.
Also, since $x > 1$, both denominators are positive:
- $1+x > 0$
- $x-1 > 0$
To compare the fractions $\frac{x}{1+x}$ and $\frac{x}{x-1}$, we compare their denominators, $1+x$ and $x-1$.
Consider the difference between the denominators:
$(1+x) - (x-1) = 1 + x - x + 1 = 2$
Since the difference is positive ($2 > 0$), we have:
$1+x > x-1$
When comparing two fractions with the same positive numerator, the fraction with the smaller denominator is larger.
Because $x-1 < 1+x$, it follows that:
$\frac{x}{x-1} > \frac{x}{1+x}$
Therefore, $F > E$, or equivalently, $E < F$.
Verify with a Value
Let's test with a value $x > 1$, for example, $x=2$.
- $E = \frac{2}{1+2} = \frac{2}{3}$
- $F = \frac{-2}{1-2} = \frac{-2}{-1} = 2$
Comparing the values: $\frac{2}{3} < 2$, which means $E < F$. This confirms our result.
Important Questions from Functions Comparing
- Weight of a person can be expressed as a function of their age. The function usually varies from person to person. Suppose this function is identical for two brothers, and it monotonically increases till the age of 50 years and then it monotonically decreases. Let $a_1$ and $a_2$ (in years) denote the ages of the brothers and $a_1 < a_2$.
Which one of the following statements is correct about their age on the day when they attain the same weight? - The ceiling function of a real number x, denoted by ce(x), is defined as the smallest integer that is greater than or equal to x. Similarly, the floor function, denoted by fl(x), is defined as the largest integer that is smaller than or equal to x. Which one of the following statements is NOT correct for all possible values of x?
- The ceiling function of a real number x, denoted by $ce(x)$, is defined as the smallest integer that is greater than or equal to x. Similarly, the floor function, denoted by $fl(x)$, is defined as the largest integer that is smaller than or equal to x. Which one of the following statements is NOT correct for all possible values of x?
