1) Use a graphing utility to obtain the graph of the function. Give the domain of the function and identify any vertical or horizontal asymptotes. x+4/ x^2+x-6.
2) Use a graphing utility to graph the function and determine the x-intercepts and
B. set y=0 and solve the resulting equation to confirm your result in part (a). y=20(2/x+1 - 3/x).
F(x) = x2 + 10x between x = 0 and x = 6
To obtain the graph of a function and determine its domain and asymptotes using a graphing utility, follow these steps:
1) Start by entering the function into the graphing utility. In this case, the first function is f(x) = (x + 4) / (x^2 + x - 6).
2) After entering the function, plot the graph on the graphing utility. The graphing utility will display the curve representing the function.
3) To determine the domain of the function, observe the x-values for which the function is defined. In this case, since there are no restrictions on the domain, the function is defined for all real numbers except where the denominator (x^2 + x - 6) is equal to zero. Therefore, the domain is all real numbers except the x-values that make the denominator zero.
4) To find any vertical asymptotes, look for values of x that make the denominator equal to zero. In this case, we need to solve the equation x^2 + x - 6 = 0. Factoring or applying the quadratic formula, we find that the solutions are x = -3 and x = 2. Hence, the vertical asymptotes occur at x = -3 and x = 2.
5) Next, let's take a look at the second function, g(x) = 20 * ((2/x) + 1 - (3/x)).
6) Enter the function g(x) into the graphing utility and plot the graph.
7) To determine the x-intercepts of the function, look for points on the graph where the y-value is zero (y = 0). These points correspond to the x-intercepts. In the graphing utility, you can use the cursor or the "zero" function to find these x-intercepts. Make sure to check both sides of the x-axis for any x-intercepts.
8) Additionally, you can check the x-intercepts by setting y = 0 in the function equation and solving for x algebraically. In this case, we solve 20 * (2/x + 1 - 3/x) = 0. Simplifying, we have 2/x - 3/x = -1. Combining like terms, we get -1/x = -1. Solving for x, we find that x = 1. Hence, the x-intercept is confirmed to be x = 1.
By following these steps and using a graphing utility, you can obtain the graph of a function, identify its domain, and determine any asymptotes, as well as find x-intercepts and confirm them algebraically.