Exam 1: Functions and Models

Medical professionals sometimes use iodine-131, a radioactive substance, to diagnose certain conditions of the thyroid gland. The formula for the proportion P of iodine-131 remaining in a patient's system t days after receiving the substance is given by P $= e ^ { ( \ln ( 1 / 2 ) / 8 ) t }$ .(a) Find the inverse of this function and explain its meaning.(b) How long does it take for the proportion to drop to 10% of the original dosage?

For what value of x is 3 $4 - x$ = $\sqrt { 3 }$ ?

Given the graph of y = 2 $x$ , find an equation of the graph that results from reflecting the given graph about (a) the line x = 0.(b) the line y = 3.(c) the line y = -1.(d) the line x = 2.

Solve the equation $e ^ { 2 x - 2 } = 4$ .

Relative to the graph of y = x $2$ $+$ 2, the graph of y = 4x $2$ $+$ 2 is changed in what way?

Find the value of $\log _ { 16 } 8$ .

Find functions f and g such that F(x) = 1 - 2 cos $2$ x = $( f \circ g ) ( x )$

Determine an appropriate viewing window for each of the following functions and use it to draw the graph.(a) f (x) = x $4$ - 6x $2$ (b) f (x) = $\frac { 1 } { x ^ { 4 } - 1 }$ (c) f (x) = sin (x $2$ ) (d) f (x) = 80x - 5x $3$ (e) f (x) = - 3x $2$ + 288x - 6862 (f) f (x) = - 3x $2$ + 288x + 6862

Solve for ln $x ^ { 2 } = [ \ln x ] ^ { 2 }$ .

Express 2 log x - 5 log (x - 10) as a single logarithm.

If a function f has an inverse and is an increasing function, can you determine if the inverse is increasing or decreasing? Explain.

Match each set of function values in the table with the formula which best fits it. Assume that a, b, and c are constants. - Function A Function B Function C x f(x) 4 340 5 367 6 400 7 428 8 463 x g(x) 4 351 5 365 6 380 7 395 8 411 x h(x) 5 322 6 354 7 390 8 429 9 472 Formula 1 Formula 2 Formula 3 =a(1.04 =b(1.08 =c(1.10

Consider the family of curves given by y = x $2$ + cx. Graph the function for values of c = - 4, - 2, 0, 2, and 4. What characteristics are shared by all of the graphs? How does changing the value of c affect the graph?

Determine the number of real solutions of the equation 5x $4$ - 3x $2$ = 10x $5$ - 20x $3$ + 5.

Given the function tan x with domain $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ find the domain of its inverse.

Find the value of log $1 / 2$ 1.

Refer to your models from Problems 6 and 7 above. Why do the two data sets produce such different models?

Find the value of $\log _ { 2 } e - \log _ { 2 } ( e / 16 )$ .

Use the graphs of f and g given below to estimate the values of f(g(x)) for x = $- 3$ , $- 2$ , $- 1$ , 0, 1, 2, and 3, and use these values to sketch a graph of y = f(g(x)).

Let f(x) = 3x - 2 and g(x) = 2 - 3x. Find the value of (f ^$\circ$ g)(x) when x = 3..