Understanding the Function f(x) 63x / (1 - 2x): Domain and Range Analysis

This article will explore the domain and range of the function f(x) 63x / (1 - 2x). We will analyze this rational function, understand its domain and range, and provide a detailed explanation to help students and professionals alike.

Introduction to the Function

The function f(x) 63x / (1 - 2x) is a rational function, which is a ratio of two polynomials. Rational functions are fascinating because they can exhibit various behaviors such as asymptotes and discontinuities, and we will delve into these aspects to fully understand the function's properties.

Domain Analysis

The domain of the function is the set of all real numbers x for which the function is defined. For a rational function like this, the domain excludes the values of x that make the denominator zero, as division by zero is undefined.

Let's analyze the denominator: 1 - 2x. Setting this to zero gives us the value of x where the function is undefined.

Step 1: Solve for the Undefined Point

[1 - 2x 0] Solving this equation, we find:

x 1/2

Thus, the domain of this function is all real numbers except x 1/2. We can write the domain as:

Domain {x ∈ ? | x ≠ 1/2}

Range Analysis

The range of a function is the set of values that the function can actually take on. For a rational function, the range can be determined by analyzing the horizontal and vertical asymptotes as well as the behavior of the function at different points.

Vertical Asymptote

The vertical asymptote is where the function approaches infinity or negative infinity as x approaches a certain value. In our function, the denominator 1 - 2x is zero at x 1/2, indicating a vertical asymptote at x 1/2. Therefore, the vertical asymptote is:

x 1/2

Horizontal Asymptote

To find the horizontal asymptote, we need to determine the behavior of the function as x approaches positive or negative infinity. For our function, the degrees of the numerator and the denominator are both 1. In this case, the horizontal asymptote is found by taking the ratio of the leading coefficients of the numerator and the denominator.

The leading coefficient of the numerator is 63, and the leading coefficient of the denominator is -2. Therefore, the horizontal asymptote is:

y 63 / -2 -3/2

Thus, the horizontal asymptote is:

y -3/2

Range Verification

Having determined the horizontal asymptote, we should verify if there are any values in the range that cannot be achieved. In this case, we need to check if the value y -3/2 can be achieved by the function.

Let's assume there is a value of x such that f(x) -3/2:

-3/2 63x / (1 - 2x)

Multiplying both sides by (1 - 2x) gives:

-3/2 * (1 - 2x) 63x

-3/2 3x 63x

63x - 3x -3/2

6 -3/2

x -3 / (2 * 60) -1/40

However, when substituting x -1/40 into the original function:

f(-1/40) 63(-1/40) / [1 - 2(-1/40)] -63/40 / [1 1/20] -63/40 / 41/20 -63/40 * 20/41 -63/82 ≠ -3/2

Thus, the value y -3/2 cannot be achieved by the function. Therefore, the range is:

Range {y ∈ ? | y ≠ -3/2}

Conclusion

Understanding the domain and range of the function f(x) 63x / (1 - 2x) is crucial for analyzing its behavior. The domain is all real numbers except x 1/2, and the range is all real numbers except y -3/2.

By grasping these key aspects, one can better comprehend the nature of rational functions and their applications in various mathematical and practical scenarios.