Sketch the graphs of f(x)=3overx+1 and g(x)=-2x-4
To sketch the graphs of the functions f(x) = 3/(x + 1) and g(x) = -2x - 4, we can follow these steps:
Step 1: Determine the domain and vertical asymptotes of f(x).
- The function f(x) = 3/(x + 1) is defined for all values of x except when x = -1, as the denominator cannot be zero.
- Therefore, the domain of f(x) is all real numbers except -1.
- Additionally, we can find the vertical asymptote by setting the denominator to zero: x + 1 = 0. Solving this equation, we find x = -1.
- This means that the graph of f(x) has a vertical asymptote at x = -1.
Step 2: Determine the x-intercept and y-intercept of f(x).
- To find the x-intercept, set f(x) = 0 and solve for x: 3/(x + 1) = 0.
- Multiplying both sides by (x + 1), we get 3 = 0.
- This equation has no solution, which means that f(x) does not have an x-intercept.
- To find the y-intercept, set x = 0 and evaluate f(x): f(0) = 3/(0 + 1) = 3.
- Therefore, the y-intercept of f(x) is (0, 3).
Step 3: Plot additional points to sketch the graph of f(x).
- Choose some values for x and find the corresponding y-values to plot more points.
- For example, when x = -2, f(x) = 3/(-2 + 1) = -3.
- Another point can be for x = 1, f(x) = 3/(1 + 1) = 3/2 = 1.5.
- By selecting a few more x-values and calculating their corresponding y-values, we can plot more points to sketch the graph of f(x).
Step 4: Sketch the graph of f(x).
- Plot the determined points on a coordinate grid and connect them smoothly to form the graph respecting the behavior around the asymptote and the points.
- Since f(x) has a vertical asymptote at x = -1, its graph should approach the asymptote but never touch or cross it.
Now let's move on to sketching the graph of g(x) = -2x -4.
Step 1: Determine the domain and range of g(x).
- The function g(x) = -2x - 4 is defined for all real numbers, so its domain is all real numbers (-∞, ∞).
- The range of g(x) is also all real numbers (-∞, ∞).
Step 2: Determine the x-intercept and y-intercept of g(x).
- To find the x-intercept, set g(x) = 0 and solve for x: -2x - 4 = 0.
- Adding 4 to both sides and then dividing by -2, we get x = -2.
- Therefore, the x-intercept of g(x) is (-2, 0).
- To find the y-intercept, set x = 0 and evaluate g(x): g(0) = -2(0) - 4 = -4.
- Therefore, the y-intercept of g(x) is (0, -4).
Step 3: Plot additional points to sketch the graph of g(x).
- Choose some values for x and find the corresponding y-values to plot more points.
- For example, when x = 1, g(x) = -2(1) - 4 = -6.
- Another point can be for x = -3, g(x) = -2(-3) - 4 = 2.
- By selecting a few more x-values and calculating their corresponding y-values, we can plot more points to sketch the graph of g(x).
Step 4: Sketch the graph of g(x).
- Plot the determined points on a coordinate grid and connect them smoothly to form the graph of g(x).
- Since g(x) is a linear function, its graph will be a straight line.