Checking Your Skills:
1. f(x) = x / (x^2 - 4)
2. g(x) = (2x + 5) / (3x + 2)
3. h(x) = (x^2 - 9) / (x + 2)
Explanation:
X-intercept: The x-intercept is the point where the graph of the function crosses the x-axis. This happens when the y-value of the function is 0. To find the x-intercept, set y to 0 and solve for x.
Y-intercept: The y-intercept is the point where the graph of the function crosses the y-axis. This happens when the x-value of the function is 0. To find the y-intercept, set x to 0 and solve for y.
Asymptote: An asymptote is a line that the graph of a function gets closer and closer to as the input value approaches a certain point, but never actually touches.
Hole: A hole in the graph of a function is a point where the function is undefined, but the graph is not continuous.
Solution:
1. f(x) = x / (x^2 - 4)
X-intercept:
0 = x / (x^2 - 4)
0 = x
x = 0
Y-intercept:
y = x / (0^2 - 4)
y = x / -4
y = 0
Asymptotes:
x^2 - 4 = 0
(x - 2)(x + 2) = 0
x = 2 or x = -2
The vertical asymptotes are at x = 2 and x = -2.
Holes: None
2. g(x) = (2x + 5) / (3x + 2)
X-intercept:
0 = (2x + 5) / (3x + 2)
0 = 2x + 5
x = -2.5
Y-intercept:
y = (2(0) + 5) / (3(0) + 2)
y = 5 / 2
Asymptotes:
3x + 2 = 0
x = -2/3
The vertical asymptote is at x = -2/3.
Holes: None
3. h(x) = (x^2 - 9) / (x + 2)
X-intercept:
0 = (x^2 - 9) / (x + 2)
0 = x^2 - 9
0 = (x - 3)(x + 3)
x = 3 or x = -3
Y-intercept:
y = (3^2 - 9) / (3 + 2)
y = 0
Asymptotes:
x + 2 = 0
x = -2
The vertical asymptote is at x = -2.
Holes: One at x = -2
To find the hole, we can factor the numerator and denominator of the function. We can see that the term (x + 2) appears in both the numerator and denominator. This means that we can cancel out these terms, which leaves us with the function f(x) = x - 3. This function is defined at x = -2, so there must be a hole at this point.
Conclusion:
Function: | X-intercept: | Y-intercept: | Asymptotes: | Holes: |
f(x) = x / (x^2 - 4) | x = 0 | y = 0 | x = 2, x = -2 | None |
g(x) = (2x + 5) / (3x + 2) | x = -2.5 | y = 5 / 2 | x = -2/3 | None |
h(x) = (x^2 - 9) / (x + 2) | x = 3, x = -3 | y = 0 | x = -2 | x = -2 |