how do uu find the x int. , y int. , domain range asytote of this equation
y=log to the base 10 (2x+3/x-5)
pleaseee need to understand before the test tmr!!!!
To find the x-intercept, we set y=0 and solve for x. However, in this case, we have a logarithmic function, so the equation becomes:
0 = log base 10 (2x + 3)/(x - 5)
We can rewrite this equation in exponential form:
10^0 = (2x + 3)/(x - 5)
Since 10^0 equals 1, we have:
1 = (2x + 3)/(x - 5)
Now, to solve for x, we cross-multiply:
(x - 5) = 2x + 3
Simplifying the equation:
x - 2x = 3 + 5
-x = 8
Multiplying both sides by -1:
x = -8
Therefore, the x-intercept is -8.
To find the y-intercept, we set x=0 and solve for y. Substituting x=0 into the original equation, we have:
y = log base 10 (2(0) + 3) / (0 - 5)
y = log base 10 (3/(-5))
Since the logarithm of a negative number is undefined, the y-intercept does not exist in this case.
To determine the domain of the function, we consider the restrictions of the logarithmic function. In this case, the only restriction is that the denominator, (x - 5), cannot be equal to zero. Thus, the domain of the function is all real numbers except x = 5.
To find the range of the function, we consider the behavior of the logarithmic function. As x approaches positive or negative infinity, the logarithm approaches negative infinity. Hence, the range of this function is all real numbers.
Regarding asymptotes, we have both vertical and horizontal asymptotes in this case. The vertical asymptote occurs when the denominator of the logarithmic function is equal to zero. In this equation, the vertical asymptote occurs at x = 5.
The horizontal asymptote can be found by considering the behavior of the logarithmic function as x approaches positive or negative infinity. In this case, the horizontal asymptote is y = 0.
To summarize,
X-intercept: -8
Y-intercept: None
Domain: All real numbers except x = 5
Range: All real numbers
Vertical asymptote: x = 5
Horizontal asymptote: y = 0