Let f, g: (1, ∞) → R$f, g: (1, \infty) \to \mathbb{R}$ be defined as f(x) = (2x + 3)/(5x + 2)$f(x) = \frac{2x + 3}{5x + 2}$ and g(x) = (2 - 3x)/(1 - x)$g(x) = \frac{2 - 3x}{1 - x}$. If the range of the function f(g(x))$f(g(x))$ on the interval [2, 4]$[2, 4]$ is [α, β]$[\alpha, \beta]$, then (1)/(β - α)$\frac{1}{\beta - \alpha}$ is equal to

Related Formula

For a quadratic equation Ax² + Bx + C = 0$Ax^2 + Bx + C = 0$ to yield real roots, its discriminant must satisfy:

Core Logic

Set y = (5 - x)/(x² - 3x + 2)$y = \frac{5 - x}{x^2 - 3x + 2}$:

Rearranging into a standard quadratic equation in terms of x$x$:

Step 1: Discriminant Method

Case I: If y = 0$y = 0$, the equation simplifies to x - 5 = 0 x = 5$x - 5 = 0 \implies x = 5$, which is a valid part of the domain. Thus, 0$0$ belongs to the range.

Case II: If y ≠ 0$y \neq 0$, for x$x$ to be real, D ≥ 0$D \ge 0$:

Step 2: Solving the Inequality

Completing the square for y² + 14y + 1 ≥ 0$y^2 + 14y + 1 \ge 0$:

This gives:

Comparing with the interval (-∞ , α ] [ β , ∞)$(-\infty , \alpha ] \cup [ \beta , \infty)$:

Step 3: Finding alpha^2 + beta^2

Using algebraic identities:

Pattern Recognition

For rational expressions of the form LinearQuadratic$\frac{\text{Linear}}{\text{Quadratic}}$, converting to a quadratic in x$x$ and forcing D ≥ 0$D \ge 0$ establishes the range boundaries elegantly.

Chapter Mix

Class 11 Mathematics: Sets, Relations and Functions

Related Formula

A function is one-one if its derivative is strictly monotonic (always positive or always negative) across its domain. It is onto if its range equals its co-domain.

Core Logic

Rewrite the function by multiplying the numerator and denominator by 2^x$2^x$:

Step 1: Check One-One property

Differentiating f(x)$f(x)$ with respect to x$x$ :

Since 22x > 0$2^{2x} > 0$ and ln 2 > 0$\ln 2 > 0$, f'(x) > 0$f'(x) > 0$ always. Thus, f(x)$f(x)$ is strictly increasing, confirming it is a one-one function.

Step 2: Check Onto property

Analyze the limits at boundaries:

Thus, the range of the function is (-1, 1)$(-1, 1)$. Since the given co-domain is (-∞, 1)$(-\infty, 1)$ and Range ≠ Co-domain$\text{Range} \neq \text{Co-domain}$ , the function is not onto.

Pattern Recognition

The expression given is a shifted form of the hyperbolic tangent function (x ln 2)$\tanh(x \ln 2)$. Hyperbolic tangent always maps to (-1, 1)$(-1, 1)$, making its restriction against (-∞, 1)$(-\infty, 1)$ non-surjective (not onto).

Chapter Mix

Class 12 Mathematics: Relations and Functions

Related Formula

For a vertical line x = x₀$x = x_0$ crossing a bounded region, the valid coordinates of y$y$ sit between the boundary lines intersecting that specific line vertical plane.

Core Logic

The point given is fixed at x = (11)/(2) = 5.5$x = \frac{11}{2} = 5.5$. We evaluate the values of y$y$ along this vertical line segment across each boundary edge.

Step 1: Evaluate Intersections

Substitute x = (11)/(2)$x = \frac{11}{2}$ into the three linear constraints:

• From x + y = 11$x + y = 11$:

• From x + 2y = 16$x + 2y = 16$:

• From 2x + 3y = 29$2x + 3y = 29$:

Step 2: Define Extrema and Multiply

Checking the internal region of the $\triangle$ bounded by these lines, the valid range for α$\alpha$ along the section line is delimited by y = 5.5$y = 5.5$ and y = 6$y = 6$:

The product of the limits is :

Pattern Recognition

Instead of drawing full coordinate diagrams or computing all three vertex points, evaluating values directly at the fixed coordinate constraint x = 5.5$x = 5.5$ saves time during multi-line area problems.

Chapter Mix

Class 11 Mathematics: Linear Inequalities Class 11 Mathematics: Straight Lines

Related Formula

Fundamental Counting Principle: If an operation can be performed in n₁$n_1$ ways, followed by a second operation in n₂$n_2$ ways, the total configurations equal n₁ × n₂$n_1 \times n_2$.

Core Logic

The domain set contains integers from 1 to 100. We divide the mapping requirements across distinct subsets of this domain.

Step 1: Choose the single element from 1, 2, , 98$\{1, 2, \dots, 98\}$

We must assign the image value 1 to exactly one positive integer from the subset 1, 2, , 98$\{1, 2, \dots, 98\}$. The number of ways to pick this single element is :

Step 2: Mapping remaining elements

The remaining 97 elements in the 1, 2, , 98$\{1, 2, \dots, 98\}$ subset cannot map to 1, so they must map to 0. This leaves exactly 1 choice per remaining element.

For the final two elements in the domain, 99 and 100, there are no structural constraints:

• Element 99 can map to either 0 or 1 (2 options) .
• Element 100 can map to either 0 or 1 (2 options).

Step 3: Total functions combination

Multiply the independent choices together :

Pattern Recognition

Separate domains tightly into restricted blocks vs completely free components. Realizing that elements 99 and 100 behave independently with full co-domain targets leaves a clear product formulation.

Chapter Mix

Class 12 Mathematics: Relations and Functions Class 11 Mathematics: Permutations and Combinations

Related Formula

Many finite fractional sum questions involving functional terms rely on identifying an underlying symmetric summation invariant, typically of the form f(x) + f(k-x) = constant$f(x) + f(k-x) = \text{constant}$.

Core Logic

First simplify the expression for f(x)$f(x)$ algebraically:

Factor out 4$4$ from the numerator and 2$2$ from the denominator:

Step 1: Establish Symmetry Pairings

Let's check the value of f(x) + f(4-x)$f(x) + f(4-x)$:

Now compute the sum directly:

Hence, whenever two input arguments sum up to 4$4$, the sum of their functional values is exactly (1)/(2)$\frac{1}{2}$.

Step 2: Group the Finite Series Terms

Consider the terms inside the requested sequence:

This complementary pairing continues up to:

This yields exactly 29$29$ distinct pairs. The single middle term left unpaired corresponds to:

Step 3: Evaluate Final Expression

Compute the total value by multiplying the grouped sum by 8:

Pattern Recognition

Whenever a symmetric set of arguments is presented inside a summation matching (k)/(n) + (N-k)/(n) = constant$\frac{k}{n} + \frac{N-k}{n} = \text{constant}$, look for an algebraic reduction of f(x)$f(x)$ that yields a uniform constant sum for symmetric pairs.

Chapter Mix

Class 11 Mathematics: Functions