+4473 For g(x)=4^x, notice that we can say y = g(x) = 4^x or y is a function of x. The same goes for y = f(x) = log(5)x.
For g(x)=4^x: To find the x-intercept, we need to set y = 0 --> y = g(x) = 0 = 4^x or simply, 0 = 4^x. Now, if we plug in any values not includng 0 for x such as x = -5, -4, -3, -1, 1, 2, etc. we get y-values that are NOT 0. Therefore, the x-intercept does not exist. If we graph g(x)=4^x, we notice that as the x-values reach negative infinity, the y-values are closer and closer to the 0 value but never 0. Therefore, there is no x-intercept.
To find the y-intercept, we need to set x = 0 --> y= g(x) = 4^0 = 1. Any number to a power of 0 is equal to 1. Now, notice that if we tried plugging in x = 0 first when we were finding the x-intercept, we would have already found the y-intercept without noticing! Therefore, the y-intercept = 1 at the point (0,1).
f(x)=log(5)x: To find the x-intercept, we need to set y = 0 --> y = f(x) = 0 = log(5)x. Since this is a log function, it is best to observe the behavior of the graph since plugging in points may be a bit too tedious. It is best to use the properties of logs to find the intercept(s). 0 = log(5)x. The log, in this case, is in the log based 10 form (log 10). So by properties of logs, we can do the following: 10^0 = 10^(log10(5)x) --> 1 = 5x [10 to the power of log based 10 cancels out leaving the 5x] --> Now, the algebra is simple and we divide both sides by 5 to obtain x = 1/5 = 0.2. However, by looking at the graph, we notice that the graph is symmteric. Therefore, the x-intercepts would be at 0.2 and -0.2 at the points (0.2, 0) & (-0.2, 0).
To find the y-intercept, we need to set x = 0 --> y = f(x) = log(5)0 = log(0). Notice that we have y = log 0 and there is no x anywhere which means there is no y-intercept since x = 0 becomes log 0 which is a continous function when graphed, the y-value goes to negative infinity because of the nature of the log 0 graph. There is no y-intercept.
