Rewriting
Expressions
Expressions
Simplify \frac{x^2-9}{x+3} .
x-3
The vertical asymptote of f(x)=\frac{3}{x+4}
x=-4
Solve \frac{6}{x}=2
x=3
An equation that is true for every allowed value of the variable is called a(n) ____________________.
An identity
In the flight model T(w)=\frac{300}{530-w} (a 300-mile flight with a planned speed of 530 mph), find T(110) and explain what it means.
T(110)=\frac{300}{420}=5/7 hours, about 43 minutes: the flight time with a 110 mph headwind.
Simplify \frac{x^2+5x+6}{x^2-4}
\frac{x+3}{x-2}
The horizontal asymptote of f(x)=\frac{4x^2+1}{2x^2-3}
y=4
Solve \frac{1}{x}+\frac{1}{2}=\frac{3}{4}
x=4 x^3 - 3x^2 + 2x
Is (x+1)^2=x^2+1 an identity? Name a value that shows why or why not.
No. For example, x = 1 gives 4 on the left and 2 on the right.
The vertical asymptote of T(w)=\frac{300}{530-w}
w=530
Divide & simplify: \frac{x^2-x-6}{x^2+x}\div\frac{x-3}{x+1}
\frac{x+2}{x}
The x- and y-intercepts of f(x)=\frac{x-6}{x+2}
(6,0) and (0,-3)
Solve \frac{x}{x-2}=2+\frac{2}{x-2}
No solution. Clearing denominators gives x = 2, but that value makes a denominator zero, so it is extraneous.
Is \frac{x^2-9}{x-3}=x+3 an identity? Explain.
No. The two sides match for every x except x = 3, where the left side is undefined.
In this flight model, this is what the vertical asymptote at w=530 means.
A headwind equal to the planned speed means the plane makes no progress, so the flight time grows without bound.
Rewrite: \frac{2x^2+7x+9}{x+2} as q+\frac{r}{d}
2x+3+\frac{3}{x+2}
The vertical and horizontal asymptotes of r(x)=2+\frac{x+1}{x-3}
x=3 and y=3
Solve \frac{2}{x+1}=\frac{x}{x+3}
x=3 or x=-2
Use the finite geometric sum formula to evaluate 1+2+4+...+2^9 .
\frac{1-2^10}{1-2}=1023
A square prism has volume 125 cubic inches and base side length w. Write the surface area S as a function of w, using S=2w^2+4wh and V=w^2h.
S(w)=2w^2+\frac{500}{w}
Write \frac{x}{x-2}-\frac{2}{x+2} as a single fraction
\frac{x^2+4}{x^2-4}
Write an equation for 1/x translated 4 units left and 2 units up, as a single fraction, and name its asymptotes
\frac{2x+9}{x+4} with asymptotes x=-4 and y=2
Solve \frac{x}{x+2}+\frac{2}{x-2}=\frac{8}{x^2-4}
No solution. Clearing denominators gives x^2=4 , but both x = 2 and x = -2 make a denominator zero.
Write 3+9+27+...+3^15 as a single fraction and evaluate it.
\frac{3(1-3^15)}{1-3}=21,523,359
Using a table or graph of S(w)=2w^2+\frac{500}{w}, find the side length that gives the least surface area, and state that least surface area.
w=5 inches, giving the least surface area of 150 square inches.