Test 1
2005, Fall

Version A

Mixture. 6 points.

We have 3 kg 40 % concentrated solution. How much of 60 % solution should be added to increase its concentration to  45% ?

Algebraic rules. 10 points.

Simplify the following expressions:

(1/5)-(1/4),		(((y^(-2))(x^4))/((y^3)(x^(-3)))), 	(1/(a+b))+ (1/((a^2)-(b^2))).

Logarithmic rules. 8 points.

Evaluate the following expressions:

((1/81)^(1/4)), 	(Subscript[log, 2]) (1/32), 	(lg10^3), 		(Subscript[lg, 1/3])27.

Elementary functions. 12 points.

Give the domains and sketch the graphs of the following functions:

(3^(-x)), 	(1/(x-3)), 	(Subscript[log, 1/3])x,	ctgx.

Straight line. 10 points.

A straight line intercepts the y-axis at y=-6, and the x-axis at x=2. Give the equation of the straight line and draw its graph.

Parabola. 12 points.

Find the peak-point equation and the factorized equation of the following quadratic function and draw its graph (with peak-point & intercepts):

y=-(x^2)-4x-3

Transformations. 10 points.

Which transformation steps on the graph of the function ((((1/2)))^x)  are required to get the graph of f(x)=1-((((1/2)))^(x-1)) ?  Make a sketch of the steps!

Logarithmic coordinate system. 10 points.

Make a (semi-)logarithmic plot of the following function (plot as a straight line): y=0.25×(8^x).

Domain of a function. 10 points.

Determine the domain of definition for the function.

f(x)=((-(x^2)+x)^(1/4))

Inverse function. 12 points.

Find and plot the inverse function of:

y=f(x)=(1/2)((x-1)^(1/3))+1