Precalc

Back in the Day

100

If f(x) is the inverse of g(x), then what is f(g(x))? Explain.

f(g(x)) = x

g(x) will map the x-value to the range, but f(x) will map this corresponding y-value back to the original x-value.

200

Graph the following rational function. Identify all asymptotes, holes, and intercepts:

f(x)= (2x^2 - 8x - 10)/ (x^2 - 25)

x-int: (-1,0)

y-int: (0,2/5)

HA: y=2

VA: x=-5

Hole: x=5 (5,1.2)

Polynomials

100

Write the equation for the following polynomial. (See graph)

y=1/5(x+1)^2(x+2)

200

Graph by hand: 3x³+4x²+x-2, given x=2/3 is a root. 

Roots at x=-1 (multiplicity 2) and x=2/3

300

Use polynomial long division to simplify: (see board)

4x-10 R16x+19

Rationals

100

Solve for all values of x:

1/x+3 + 14/x^2-9 = 1

x=5 and x=-4

200

Solve for x: 1/x + 1/x+5 = 5

x=0.208 and x=-.481

300

In still water, Bogdan canoes at a pace of 8 kilometers an hour. In the river, it takes him the same amount of time to go upstream 5 km as it does to go downstream 10 km. Calculate the speed of the river (r). 

r = 2.7km

Functions

100

A function and its inverse are reflected over the line:

y=x

200

Find the inverse of f(x)= 2x+6 / x-1

f^-1(x) = 6+x / x-2

300

Find f(g(x)), given f(x) = 3x-7 / x-5 and g(x) = 5x-7 / x-2

f(g(x)) = x

Exponentials

100

Solve for x:

5^2x=25^x+3

x=6

200

Write an exponential equation that goes through the coordinates (0,10) and (8,3)

y=10 * 0.86^x

300

Solve for x: 

9^{x^2 + 0.5}= 1/27^x

x=-1/2 and x=-1