Finding Ranges Of Rational Functions And Y-Intercepts Of Polynomials

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22. Finding the Range of the Rational Function f(x) = (x-2)/(x-3)

Determining the range of a function is a fundamental concept in mathematics, particularly when dealing with rational functions. In this section, we will delve into the process of finding the range of the function f(x) = (x-2)/(x-3). Understanding the behavior of rational functions, including their asymptotes and discontinuities, is crucial for accurately identifying the set of all possible output values. To find the range, we must consider the horizontal asymptote, which indicates the value that the function approaches as x tends to positive or negative infinity. We must also account for any values that f(x) cannot take due to vertical asymptotes or holes in the graph. This comprehensive analysis will help us pinpoint the range of the given function and choose the correct option from the provided choices.

Analyzing the Function

To find the range of the function f(x) = (x-2)/(x-3), we start by recognizing that this is a rational function. Rational functions are defined as the ratio of two polynomials. In this case, both the numerator (x-2) and the denominator (x-3) are linear polynomials. The first step in analyzing the range is to identify any vertical asymptotes. These occur where the denominator is equal to zero. Setting x-3 = 0, we find that there is a vertical asymptote at x = 3. This means that the function is undefined at x = 3, and the function's values will approach infinity (or negative infinity) as x gets closer to 3. Next, we need to consider the horizontal asymptote. The horizontal asymptote describes the function's behavior as x approaches positive or negative infinity. For rational functions, the horizontal asymptote can be found by comparing the degrees of the polynomials in the numerator and the denominator. Here, both polynomials are of degree 1 (linear). When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. In this case, the leading coefficient of both the numerator and the denominator is 1, so the horizontal asymptote is y = 1/1 = 1. This means that as x becomes very large (positive or negative), the function f(x) approaches the value 1.

Determining the Range

The presence of a horizontal asymptote at y = 1 indicates that the function f(x) will get arbitrarily close to 1 but will never actually equal 1, unless there is a value of x for which f(x) = 1. To check this, we set f(x) = 1 and solve for x:

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

Multiplying both sides by (x-3), we get:

x - 3 = x - 2

Subtracting x from both sides, we are left with:

-3 = -2

This is a contradiction, which means that there is no value of x for which f(x) = 1. Therefore, the function never takes the value 1. The range of the function is all real numbers except for 1. This can be written in set notation as R {1}, where R represents the set of all real numbers and {1} indicates that 1 is excluded from the set. Considering the given options:

  • A. R {0, 1} - Incorrect, as 0 is in the range.
  • B. R - Incorrect, as 1 is not in the range.
  • C. R {2, 3} - Incorrect, as this refers to the domain restrictions rather than the range.
  • D. R {1} - Correct, as it excludes only 1 from the set of real numbers.

Therefore, the correct answer is D. R {1}.

Conclusion

The process of finding the range of a rational function involves identifying vertical and horizontal asymptotes and understanding the function's behavior as x approaches infinity. By setting f(x) = 1 and showing that there is no solution, we confirmed that 1 is excluded from the range. This systematic approach allows us to accurately determine the set of all possible output values for the given function, leading us to the correct answer. Understanding these concepts is vital for further studies in calculus and advanced mathematical analysis.

23. Determining the Y-Intercept of the Polynomial Function y = ax³ + bx² + cx + d

Finding the y-intercept of a polynomial function is a straightforward yet crucial step in understanding the graph and behavior of the function. The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when the x-coordinate is zero. In this section, we will explore how to determine the y-intercept of the polynomial function y = ax³ + bx² + cx + d. This involves substituting x = 0 into the function and solving for y. The resulting value of y gives the y-coordinate of the y-intercept, which is a key point for graphing and analyzing polynomial functions.

Understanding the Concept of Y-Intercept

The y-intercept of a function is the point where the graph of the function intersects the y-axis. This intersection occurs when the x-coordinate is equal to 0. In other words, to find the y-intercept, we need to determine the value of y when x = 0. For a polynomial function, this is a particularly simple process because we only need to substitute 0 for x in the function's equation. This concept is fundamental in graphing and understanding the behavior of various functions, not just polynomials, but also trigonometric, exponential, and logarithmic functions. The y-intercept provides a critical anchor point on the graph, making it easier to sketch the curve and visualize the function's behavior.

Finding the Y-Intercept

To find the y-intercept of the polynomial function y = ax³ + bx² + cx + d, we substitute x = 0 into the equation:

y = a(0)³ + b(0)² + c(0) + d

Simplifying the equation, we get:

y = a(0) + b(0) + c(0) + d

y = 0 + 0 + 0 + d

y = d

This result shows that the y-intercept occurs when y = d. Therefore, the coordinates of the y-intercept are (0, d). This means that the graph of the polynomial function y = ax³ + bx² + cx + d intersects the y-axis at the point (0, d). In practical terms, the constant term d in the polynomial directly gives us the y-coordinate of the y-intercept. This makes it easy to identify the y-intercept by simply looking at the equation of the polynomial.

Analyzing the Options

The question provides options for the y-intercept, and we need to identify the correct one. Based on our derivation, the y-intercept is the point where x = 0, and the corresponding y-value is d. Therefore, the correct representation of the y-intercept is the point (0, d). Let's analyze the given options:

  • A. (y(0), 0) - This option is incorrect because it suggests that the y-coordinate of the y-intercept is 0, and the x-coordinate is y(0), which is the y-value when x = 0. However, the correct format should be (x, y), and we know that x = 0 at the y-intercept.
  • B. (0, y(0)) - This option is the correct representation of the y-intercept. Here, x = 0, and y(0) represents the y-value when x = 0, which we found to be d. Thus, this option matches our derived result of (0, d).
  • C. (0, d) - This option is also correct and represents the y-intercept as the point where x = 0 and y = d. This is consistent with our derivation.
  • D. (d, 0) - This option is incorrect because it represents the x-intercept (where the graph intersects the x-axis) rather than the y-intercept. At the x-intercept, y = 0, not x.

Conclusion

In conclusion, the y-intercept of the polynomial function y = ax³ + bx² + cx + d is the point (0, d). This is found by substituting x = 0 into the equation and solving for y. The constant term d in the polynomial directly corresponds to the y-coordinate of the y-intercept. Understanding how to find the y-intercept is crucial for graphing polynomials and analyzing their behavior. It provides a simple yet powerful way to identify a key point on the graph of the function. This fundamental concept is widely used in algebra, calculus, and various applications of mathematics.