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Michael A.
Math Tutor with magna cum laude B.A. degree in Mathematics
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Pre-Calculus
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Question:

Find, if they exist, the aysmptotes, intercepts, and domain of the function: g(x) = (4x - 8)/(x - 3)

Michael A.

The vertical asymptote of a rational function occurs where the function is undefined, which is when the denominator is equal to zero. For the above function, when x = 3, the denominator is equal to 0, and division by zero is undefined. So, g(x) has a vertical asymptote at x = 3. This also gives us the domain of g(x). The function is defined for all real values of x except for x = 3. To find the horizontal asymptotes of a rational function, we divide each term in the numerator by the highest power of x in the function. In this case, the highest power of x is x to the first power. Dividing the leading term in the numerator by x gives us 4, and in the denominator it gives us 1. The other terms in the function will tend to zero as x grows larger and larger, or approaches infinity. Therefore, g(x) tends toward 4/1 = 4 as x approaches infinity. This is the same as taking the limit of the function. Consequently, g(x) has a horizontal asymptote at y = 4. Finally, to find the x-intercepts we let y = 0. The only time that g(x) will equal zero is when the numerator is equal to 0. We need to solve the following equation: 4x - 8 = 0 Isolate the variable term by adding 8 to both sides. 4x = 8 Divide both sides of the equation by the coefficient of x, which is 4. x = 2 This tells us that g(x) has an x-intercept at the coordinate (2, 0). To find the y-intercepts, we let x = 0. g(0) = [4(0) - 8]/(0 - 3) = (0 - 8)/(-3) = -8/-3 = 8/3 Hence, g(x) has a y-intercept at the coordinate (0, 8/3).

Pre-Algebra
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Question:

Change the listed percent to a fraction, then a whole number or a reduced mixed number. 375%

Michael A.

All percentages are based on 100, so we can write any percent as a fraction by placing it over 100. So 375% is equivalent to: 375/100, which can be reduced by dividing the numerator and denominator by the greatest common factor(GCF), which is 25. 375/100 = 15/4 Since 4 does not divide evenly into 15, this improper fraction can be written as a mixed number by dividing the denominator into the numerator. The number 4 divides evenly into 15 a total of 3 times with a remainder of 3. Therefore, 15/4 can be written as the mixed number 3 3/4.

Algebra
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Question:

What is the solution to the system of equations show below? 3x + y = 11 y = x + 3

Michael A.

We can use the Substitution Method to solve the system of equations. If we replace y in the first equation with (x + 3), we get: 3x + (x + 3) = 11 Combine like terms to get: 4x + 3 = 11 Isolate the variable term by subtracting 3 from both sides of the equation. 4x = 8 Divide both sides of the equation by the coefficient of x to solve for x. x = 2 Now, we can substitute x = 2 into either equation to solve for y. Let's choose the second equation. y = 2 + 3 = 5 The solution to this system of equations is (2, 5). This can be verified by replacing x and y into the first equation with 2 and 5, respectively. 3(2) + 5 = 6 + 5 = 11

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