Problem 29 - Calculus for JEE ADVANCED

Question

Draw the graph of \(y=\log _{0.5}|x|\) and \(y=2|x|\). Clearly, from the graph, there are two solutions.

Step-by-Step Solution

Answer

The solutions are the approximate points where the graphs intersect, around \(x = -1\) and \(x = 1\).

Step 1: Understand the functions

We have two functions to graph: 1) \(y = \log_{0.5}|x|\) and 2) \(y = 2|x|\). The logarithmic function uses the base 0.5. The absolute value ensures the function is defined for all non-zero \(x\). The second function is a simple V-shaped graph that is symmetric about the y-axis.

Step 2: Plot the function \(y = \log_{0.5}|x|\)

When \(x > 0\), \( |x| = x \) and when \( x < 0 \), \( |x| = -x \). The function \( y = \log_{0.5}|x| \) is decreasing as the logarithm base is less than 1. It is undefined at \( x = 0 \). As \( x \) approaches zero from either side, \( y \) tends to infinity, and as \( x \) moves away from zero, \( y \) tends toward negative infinity.

Step 3: Plot the function \(y = 2|x|\)

The function \(y = 2|x|\) is a straight line in two segments: one segment with slope 2 when \(x > 0\) and another segment with slope -2 when \(x < 0\). The graph passes through the origin (\(0,0\)) and is V-shaped.

Step 4: Identify intersections

To find the intersection points, set \( \log_{0.5}|x| = 2|x| \). This requires solving the equation by finding the \(x\) values where these graphs intersect visually. Due to symmetry, check solutions for \( x > 0\) and reflection for \( x < 0\).

Step 5: Verify intersection points graphically

By observing the graph, you can see the function of \(y = \log_{0.5}|x|\) intersects with \(y = 2|x|\) at two points. Graphically solve to find these intersections, likely around \(x \approx -1\) and \(x \approx 1\), based on the shape of both functions.

Key Concepts

logarithmic functions

Logarithmic functions are a unique type of function that involve logarithms. In the problem we explore, the function is written as \( y = \log_{0.5}|x| \). This means we use the logarithm's base, which is 0.5, to scale the absolute value of \( x \). The function is particularly interesting because the base of the logarithm is less than 1, leading to a decreasing graph. This results in a function that slopes downward as \( x \) moves away from the origin. As \( x \) moves toward zero from either direction, the function's value approaches positive infinity. Conversely, as \( x \) increases in magnitude, getting further from zero, the function trends toward negative infinity. Such characteristics make the function behave differently than typical logarithmic functions with bases greater than 1. Graphically, the function is undefined at \( x = 0 \), meaning there is a vertical asymptote at the y-axis. Understanding how these elements interact is essential when graphing or solving systems with logarithmic terms.

absolute value functions

Absolute value functions create interesting graphs because of their symmetry. The given function, \( y = 2|x| \), highlights this symmetry. It's known for forming a V-shaped graph that is symmetric about the y-axis. This shape arises because the absolute value \(|x|\) constrains all \( x \) values to non-negative results. For positive \( x \), the function behaves linearly as \( y = 2x \). For negative \( x \), it behaves as \( y = -2x \). Hence, regardless of the sign of \( x \), the output is always non-negative, giving this characteristically symmetric V-shape. The slope of 2 indicates that for every unit increase or decrease in \( x \), the \( y \) value changes by 2, further providing the steepness of the graph. The origin, where the graph intersects both axes, is a key feature since it's a turning point in the V-shape. Recognizing and graphing absolute value functions efficiently perfects one's ability in function transformation and symmetry.

graph intersections

Finding graph intersections involves determining at which points different functions cross each other. In this exercise, we are interested in the intersection points of the graphs \( y = \log_{0.5}|x| \) and \( y = 2|x| \). Intersections are found by setting the equations equal to one another: \( \log_{0.5}|x| = 2|x| \). Solving this graphically often reveals key intersections more clearly than algebraically due to the complexity of combining logarithmic and absolute value functions. By plotting both graphs, you observe the behaviors: \( y = \log_{0.5}|x| \)'s steep decline and \( y = 2|x| \)'s linear V-shape. Observing these, the intersections appear symmetrically at approximately \( x = -1 \) and \( x = 1 \). The position of these intersections indicates where both functions yield the same output, hence crossing each other on the coordinate plane. Such intersection points are crucial in helping understand how and where different mathematical models compare directly.