For each function below, determine the characteristics listed and then sketch the function: (5 marks)
f(x)=2x-3/x+2
x-intercept(s):
y-intercept(s):
Vertical Asymptote(s):
Horizontal Asymptote(s):
To determine the characteristics of the function f(x) = (2x - 3)/(x + 2) and sketch the function, we need to find the following:
1. x-intercept(s): The x-intercept(s) occur when the function crosses or touches the x-axis. To find the x-intercepts, we set y = 0 and solve for x.
0 = (2x - 3)/(x + 2)
Cross multiply: 0 = 2x - 3
Solve for x: 2x = 3
x = 3/2
Therefore, the function has a single x-intercept at (3/2, 0).
2. y-intercept(s): The y-intercept(s) occur when the function crosses or touches the y-axis. To find the y-intercepts, we set x = 0 and solve for y.
y = (2(0) - 3)/(0 + 2)
Simplify: y = -3/2
Therefore, the function has a y-intercept at (0, -3/2).
3. Vertical Asymptote(s): Vertical asymptotes occur where the function goes towards infinity or negative infinity. To find vertical asymptotes, we look for values of x that make the denominator equal to zero.
x + 2 = 0
Solve for x: x = -2
Therefore, the function has a vertical asymptote at x = -2.
4. Horizontal Asymptote(s): Horizontal asymptotes indicate where the function approaches a certain value as x goes towards positive or negative infinity. To find horizontal asymptotes, we look at the highest power in the numerator and denominator.
In this case, the highest power in the numerator is 1 (x^1) and the highest power in the denominator is also 1 (x^1).
Since the powers are equal, we divide the coefficients of the highest power terms, which gives us 2/1 = 2.
Therefore, the function has a horizontal asymptote at y = 2.
Now that we have determined the characteristics of the function f(x) = (2x - 3)/(x + 2), we can sketch the function using this information. The x-intercept is at (3/2, 0), the y-intercept is at (0, -3/2), there is a vertical asymptote at x = -2, and a horizontal asymptote at y = 2.