Problem 33 - Calculus for JEE ADVANCED

Question

$$ f(x)=\frac{1}{\sqrt{4 x-\left|x^{2}-10 x+9\right|}} $$ For \(f(x)\) to be defined, \(\left|x^{2}-10 x+9\right|<4 x\) or \(x^{2}-10 x+9<4 x\) and \(x^{2}-10 x+9>-4 x\) or \(x^{2}-14 x+9<0\) and \(x^{2}-6 x+9>0\) or \(x \in(7-\sqrt{40}, 7+\sqrt{40})\) and \(x \in R-\\{3\\}\) or \(x \in(7-\sqrt{40}, 3) \cup(3,7+\sqrt{40})\)

Step-by-Step Solution

Answer

Domain is \( x \in (7-\sqrt{40}, 3) \cup (3, 7+\sqrt{40}) \).

Step 1: Understand the Problem

In order to find the domain of \( f(x) = \frac{1}{\sqrt{4x - |x^2 - 10x + 9|}} \), we need to ensure the expression inside the square root is positive. This means \( 4x - |x^2 - 10x + 9| > 0 \).

Step 2: Break Down Absolute Value

The absolute value condition \( |x^2 - 10x + 9| < 4x \) can be split into TWO conditions: \( x^2 - 10x + 9 < 4x \) and \( x^2 - 10x + 9 > -4x \).

Step 3: Solve Inequality 1

Let's solve \( x^2 - 10x + 9 < 4x \):\Rearrange: \( x^2 - 14x + 9 < 0 \).\Factor, or use the quadratic formula to find roots: roots of the quadratic are evaluated to determine intervals.

Step 4: Solve Inequality 2

For the other inequality \( x^2 - 10x + 9 > -4x \): \Rearrange: \( x^2 - 6x + 9 > 0 \).\Solve this quadratic inequality, and find where the expression is greater than zero.

Step 5: Combine the Solutions

Combine the intervals found from both inequalities to find common intervals. This means looking for intervals where both conditions are satisfied.

Step 6: Exclude Disallowed Points

Notice that \( x eq 3 \) because the quadratic \( x^2 - 10x + 9 \) has a double root at \( x = 3 \). So any interval containing 3 should exclude this value.

Step 7: Final Domain Solution

The common intervals derived from previous steps are: \( x \in (7-\sqrt{40}, 3) \cup (3, 7+\sqrt{40}) \). These are the allowed values for \( x \) such that \( f(x) \) is well-defined.

Key Concepts

Domain of a Function

The domain of a function tells us all the possible input values (usually represented as \(x\)) that a function can accept. Knowing the domain ensures that we evaluate the function only for values where the function is defined without any mathematical issues, like division by zero or negative square roots.
For a function to be defined everywhere, it should not create any mathematical impossibilities, such as taking the square root of a negative number or dividing by zero.
The given function, \(f(x) = \frac{1}{\sqrt{4x - |x^2 - 10x + 9|}}\), lives inside a square root in the denominator, meaning the expression \(4x - |x^2 - 10x + 9|\) must be greater than zero to avoid these issues. This condition ensures the square root is real and the denominator is non-zero, preventing undefined values.

Absolute Value Inequality

Absolute value inequalities involve expressions where the absolute value (the non-negative magnitude) of an expression must satisfy certain conditions. Absolute values remove negative signs, leading to non-negative results.
In an inequality like \(|x^2 - 10x + 9| < 4x\), it tells us the absolute value of \(x^2 - 10x + 9\) must be less than \(4x\).
This inequality needs to be broken into two separate inequalities to solve it effectively:

• \(x^2 - 10x + 9 < 4x\)
• \(x^2 - 10x + 9 > -4x\)

Creating these separate cases allows us to understand the range of \(x\) values for which the original inequality holds. Absolute value inequalities like this are broken into simpler linear or quadratic inequalities to make them manageable.

Quadratic Inequality

A quadratic inequality involves a quadratic expression (an expression with \(x^2\), the square of \(x\)) compared to another value. Solving these inequalities reveals intervals where the inequality holds true.
The quadratic inequalities from our problem:

• \(x^2 - 14x + 9 < 0\)
• \(x^2 - 6x + 9 > 0\)

are solved separately.
To solve, find the roots of each quadratic equation using factoring or the quadratic formula. These roots divide the number line into sections, testing each interval to see where the inequality is true.
For example, in \(x^2 - 14x + 9 < 0\), roots provide checkpoints to determine where the curve lies below the x-axis (i.e., negative). Tools like sign testing or analyzing the graph of the quadratic can help identify these intervals.

Interval Notation

Interval notation is a shorthand used to describe sets of numbers along a number line. It captures the essence of ranges clearly and succinctly.
Open intervals (like \((a, b)\)) indicate numbers between \(a\) and \(b\) not including \(a\) or \(b\). Closed intervals, expressed as \([c, d]\), include the end values \(c\) and \(d\).
For our exercise, the solution is represented in interval notation as \((7-\sqrt{40}, 3) \cup (3, 7+\sqrt{40})\). The union symbol \(\cup\) shows the values in either of the two intervals. The exclusion of \(3\) is crucial since it would cause the original expression to be undefined.
Understanding interval notation helps clearly communicate real-number solutions without ambiguity, specifying exact ranges of inclusion or exclusion needed for solutions.