Good morning @everyone. Ito ang sasagutan nyo sa Gen Math.
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Given the function, f(x) = (x-3)/ (x+3)
Find the ff:
A. equation
B. TOV (x = - 3 to 3)
C. graph
D. vertical asymptote
E. horizontal asymptote
F. domain
G. zeroes of function
H. x-intercept
I. y-intercept
Answer (1)
Answer: A. Equation: The equation of the function is already given: f(x) = (x - 3) / (x + 3) B. TOV (x = -3 to 3) TOV stands for "Table of Values". Let's create a table of values for x from -3 to 3, excluding x = -3 (since it makes the denominator zero): x f(x) = (x - 3) / (x + 3) -2 (-2 - 3) / (-2 + 3) = -5 -1 (-1 - 3) / (-1 + 3) = -2 0 (0 - 3) / (0 + 3) = -1 1 (1 - 3) / (1 + 3) = -0.5 2 (2 - 3) / (2 + 3) = -0.2 C. Graph: You can plot the points from the TOV and connect them smoothly. Here's a general description of the graph: - Shape: The graph will have a curve that approaches vertical and horizontal asymptotes.- Vertical Asymptote: There will be a vertical asymptote at x = -3 (explained below).- Horizontal Asymptote: There will be a horizontal asymptote at y = 1 (explained below). D. Vertical Asymptote: - Definition: A vertical asymptote is a vertical line that the graph approaches but never touches.- Finding it: Vertical asymptotes occur where the denominator of the function equals zero. In this case, x + 3 = 0, so x = -3. E. Horizontal Asymptote: - Definition: A horizontal asymptote is a horizontal line that the graph approaches as x goes to positive or negative infinity.- Finding it: For rational functions (fractions), compare the degrees of the numerator and denominator:- Degrees are equal: The horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator). In this case, both numerator and denominator have a degree of 1, so the horizontal asymptote is y = 1/1 = 1. F. Domain: - Definition: The domain of a function is the set of all possible input values (x-values).- Finding it: The function is undefined when the denominator is zero (x = -3). Therefore, the domain is all real numbers except -3. We can write this as: Domain: x ≠ -3 G. Zeroes of the Function: - Definition: Zeroes of a function are the x-values where the function equals zero (f(x) = 0).- Finding it: Set the numerator equal to zero and solve:x - 3 = 0x = 3- Zero of the function: x = 3 H. x-intercept: - Definition: The x-intercept is the point where the graph crosses the x-axis. This occurs when y = 0.- Finding it: We already found the zero of the function, which is the same as the x-intercept: x-intercept: (3, 0) I. y-intercept: - Definition: The y-intercept is the point where the graph crosses the y-axis. This occurs when x = 0.- Finding it: Substitute x = 0 into the function:f(0) = (0 - 3) / (0 + 3) = -1- y-intercept: (0, -1) To summarize: - Equation: f(x) = (x - 3) / (x + 3)- TOV: See the table above.- Graph: A curve approaching vertical and horizontal asymptotes.- Vertical Asymptote: x = -3- Horizontal Asymptote: y = 1- Domain: x ≠ -3- Zeroes of the function: x = 3- x-intercept: (3, 0)- y-intercept: (0, -1)
