Answer:
Function Analysis of f(x) = (x^3 - 9x)/(x^3 + x^2 - 8x - 12)
To analyze the function f(x) = (x^3 - 9x)/(x^3 + x^2 - 8x - 12), we can start by examining its properties, such as domain, range, intercepts, asymptotes, and behavior.
Domain and Range: The domain of a function is the set of all possible input values for which the function is defined. In this case, the function is defined for all real numbers except where the denominator is equal to zero. To find these values, we solve the equation x^3 + x^2 - 8x - 12 = 0. By factoring or using other methods, we can determine the values of x that make the denominator zero. The range of the function can be found by analyzing its behavior as x approaches positive or negative infinity.
Intercepts: To find the y-intercept, we can evaluate f(0). The x-intercepts are found by setting f(x) equal to zero and solving for x.
Asymptotes: The function may have horizontal, vertical, or slant asymptotes. Horizontal asymptotes occur when the degree of the numerator and denominator are equal. Vertical asymptotes occur where the denominator is equal to zero. Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator.
Behavior: We can also analyze the behavior of the function as x approaches positive or negative infinity. This involves looking at end behavior and any potential turning points or local extrema.
By thoroughly analyzing these properties, we can gain a comprehensive understanding of the behavior and characteristics of the given function.
Function Analysis: f(x) = (x^3 - 9x)/(x^3 + x^2 - 8x - 12)
Explanation:
Function Analysis of f(x) = (x^3 - 9x)/(x^3 + x^2 - 8x - 12)
To analyze the function f(x) = (x^3 - 9x)/(x^3 + x^2 - 8x - 12), we can start by examining its properties, such as domain, range, intercepts, asymptotes, and behavior.
Domain and Range: The domain of a function is the set of all possible input values for which the function is defined. In this case, the function is defined for all real numbers except where the denominator is equal to zero. To find these values, we solve the equation x^3 + x^2 - 8x - 12 = 0. By factoring or using other methods, we can determine the values of x that make the denominator zero. The range of the function can be found by analyzing its behavior as x approaches positive or negative infinity.
Intercepts: To find the y-intercept, we can evaluate f(0). The x-intercepts are found by setting f(x) equal to zero and solving for x.
Asymptotes: The function may have horizontal, vertical, or slant asymptotes. Horizontal asymptotes occur when the degree of the numerator and denominator are equal. Vertical asymptotes occur where the denominator is equal to zero. Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator.
Behavior: We can also analyze the behavior of the function as x approaches positive or negative infinity. This involves looking at end behavior and any potential turning points or local extrema.
By thoroughly analyzing these properties, we can gain a comprehensive understanding of the behavior and characteristics of the given function.
Function Analysis: f(x) = (x^3 - 9x)/(x^3 + x^2 - 8x - 12)