Analyzing the Function f(x) (x - 4) / (x - 1) : Domain, Range, and Graphical Analysis

Explore the domain, range, and graphical behavior of the real function f(x) (x - 4) / (x - 1). This analysis will help you understand the vertical and oblique asymptotes, limits at certain points, and the overall graphical representation.

1. Determining the Domain

The domain of the function f(x) is defined as all real numbers except for the value that makes the denominator zero. For the function f(x) (x - 4) / (x - 1), the denominator x - 1 equals zero when x 1. Therefore, the domain of f(x) is:

D R {1}

This means the function is defined for all real numbers except x 1.

2. Vertical Asymptotes

Since the function is undefined at x 1, this point is a vertical asymptote. The vertical asymptote occurs where the denominator is zero, which is at x 1. This can be confirmed by evaluating the limits as x approaches 1 from both the left and the right.

2.1 Left-hand Limit as x Approaches 1

As x approaches 1 from the left, the denominator x - 1 is a small negative value, making the function approach negative infinity:

limx → 1- f(x) limx → 1- (x - 4) / (x - 1) ?∞

2.2 Right-hand Limit as x Approaches 1

As x approaches 1 from the right, the denominator x - 1 is a small positive value, making the function approach positive infinity:

limx → 1 f(x) limx → 1 (x - 4) / (x - 1) ∞

3. Behavior at Infinity

Evaluating the limits as x approaches infinity helps understand how the function behaves at large values.

3.1 Limit as x Approaches Infinity

As x becomes very large, the term x - 4 becomes dominant in the numerator, and the term x - 1 in the denominator also becomes large, making the function approach a linear behavior:

limx → ∞ f(x) limx → ∞ (x - 4) / (x - 1) 1

This suggests that the function has an oblique (slant) asymptote.

4. Finding the Oblique Asymptote

To find the oblique asymptote, we need to determine the slope and the y-intercept of the asymptote.

4.1 Finding the Slope (m)

The slope can be found by dividing the function by x and taking the limit as x approaches infinity:

m limx → ∞ [f(x) / x] limx → ∞ [ (x - 4) / (x - 1) / x] 1

4.2 Finding the Y-Intercept (n)

The y-intercept can be found by taking the limit of f(x) - mx as x approaches infinity:

n limx → ∞ [f(x) - mx] limx → ∞ [(x - 4) / (x - 1) - x] -1

The equation of the oblique asymptote is then:

y x - 1

5. Range of the Function

The range of the function is all real numbers, which can be determined by considering the fact that the function has an oblique asymptote and a vertical asymptote but no horizontal asymptote. This is evident because the function does not have a maximum or minimum value, and it spans from negative to positive infinity as x varies over the domain.

Therefore, the range of f(x) is:

Im(f) R

6. Graphical Analysis

For a more visual understanding, you can graph the function using software like Scientific WorkPlace 5.5 or more advanced tools like Wolfram Alpha. However, note that some graphical representations may not include the exact oblique asymptote.

7. Intercepts of the Function

The function can also be analyzed for its intercepts. Setting f(x) 0 to find the x-intercepts:

x - 4 / (x - 1) 0 gives x^2 - x - 4 0.

Solving this quadratic equation using the quadratic formula gives:

x_1 (-(-1) sqrt((-1)^2 - 4 * 1 * (-4))) / (2 * 1) 1.561552813

x_2 (-(-1) - sqrt((-1)^2 - 4 * 1 * (-4))) / (2 * 1) -2.561552813

Thus, the intercepts of the function on the x-axis are:

x 1.561552813 and x -2.561552813.

For a detailed and faithful representation, it's recommended to use advanced mathematical software.

In conclusion, the function f(x) (x - 4) / (x - 1) has a domain of all real numbers except 1, a range of all real numbers, and an oblique asymptote at y x - 1. It also has vertical and x-intercepts, making it a comprehensive and interesting function to analyze.