A)     f _° g (x)  

Solution

        f(g(x))  =  f(x + 3)  =  3(x+ 3) + 2  =  3x + 11

B)        f(x + h) - f(x)
                                                  
                    h

Solution

        f(x + h)  =  3(x + h) + 2

so

        f(x + h) - f(x)  =  3(x + h) + 2 - (3x + 2)  =  3x + 3h + 2 - 3x - 2  =  3h

Dividing by h gives

         f(x + h) - f(x)              3h
                                    =              =  3
                  h                         h

C)   g(f(1))     

Solution

        f(1)  =  3(1) + 2  =  5

        g(f(1))  =  g(5)  =  5 + 3  =  8

D)   c _° f (2)     

Solution

        f(2)  =  3(2) + 2  =  8

        c(8)  =  -1    c is always -1

E)         c(x)g(x)
                                               
            f(x) - 7c(x)

Solution

  Plugging in, we get

             (-1)(x + 3)                -x - 3            -(x + 3)                 1
                                       =                   =                        =  -             
            3x + 2 - 7(-1)            3x + 9            3(x + 3)                3

Problem 6  You are constructing a rectangular room such that one side of the room is 14 feet longer than the other side, and the distance from opposite corners is 26 feet.  What are the dimensions of the room?  Give your answer accurate to two decimal places.

  Solution

We use the Pythagorean Theorem

        x² + (x + 14)²  =  26²

        x² + x² + 28x + 196  =  676        Multiplying out

        2x² + 28x - 480  =  0        combing like terms and bringing to the left

        x² + 14x - 240  =  0        Dividing by 2

        (x + 24)(x - 10)  =  0        factoring

        x  =  10        x cannot be negative

The dimensions are 10 x 24.

Problem 7  Graph the quadratic function.  Label any intercepts, the vertex, and the axis of symmetry.

        y  =  -2x² + 4x + 6

Solution

The x-coordinate of the vertex is 

         x  =  -b/2a  =  -4/-4  =  1

Now plug 1 into the equation to get

        y  =  -2(1)² + 4(1) + 6  =  8

Hence the vertex is at (1,8).

To find the y-intercept plug in 0 for x to get

        (0,6)

To find the x-intercept, plug in 0 for y to get

        0  =  -2x² + 4x + 6

        0  =  x² - 2x - 3        Dividing by -2

        0  =  (x - 3)(x + 1)    Factoring

        x  =  3  or   x  =  -1    The zero product rule

The x-intercepts are 

        (3,0)    and    (-1,0)

Notice that the coefficient of x² is negative, so the graph is concave down. Now plot the points and sketch the graph.

Problem 8  The graphs of y = f(x) and y = g(x) are given below.  Find

A.      f(0)

B.    g(-1)

C.     g _° f (1)  

D.         f(1)
                             
    g(-1)

  Solutions

A.  f(0) is the y-intercept of the graph of f(x) which is -1.

B.  g(-1)  =  1 since the graph goes through (-1,1).

C.  Since f(1)  =  -1, we plug -1 into g.  g(-1)  =  1.

D.  f(1)  =  -1, g(-1)  =  1.  Now divide to get -1/1  =  -1.

Problem 9  Find the domain and range of the following functions

A.  f(x)  =  2^x-1 + 3

Solution

This function is an exponential function.  Exponential functions have all real numbers as their domain.  The range of the "unshifted" exponential function is all real numbers greater than 0.  Since this function is shifted up by 3, the new range will also shift up by 3.  Thus the range is all real numbers greater than 3.  Notice that the horizontal shift "right 1" does not affect the domain or the range.

B.  f(x)  =  log₈(x+2) - 4

Solution

This function is a logarithm function.  The "unshifted logarithm functions have all numbers greater than 0 as its domain.  Since this function is shifted 2 to the left, the domain is also shifted 2 to the left.  Thus the domain is all real numbers greater than -2.  The range of logarithm function is all real numbers.  Notice that the vertical shift "down 4" does not affect the range.

Problem 10

Sketch the graph of y = 5^x.

Solution

This is an exponential function. We first find a few points.

The graph is shown below

Problem 11

Solve for w in 

        2^2w  =  1/256

Solution

First notice that 256 is a power of 2 (powers of 2 are 2, 4, 8, 16, 32, 64, 128, 256) hence

1/256  =  2⁸

256  =  2^-8

We have

2^2w  =  2^-8

so that

2w  =  -8

or

        w  =  -4

Problem 12

When a certain radioactive element decays, the amount to the element A at any time t is given by

        A  =  25 (2^t/1500)

How much of the element will be left after 3000 years?

Solution

We plug 3000 into this equation for t to get

        A  =  25 (2^3000/1500)  =  (25)2²  =  100

Problem 13

Sketch the graph of the function below.  Label the vertex and the axis of symmetry.

y = -2(x - 1)² + 5

Solution

This is a parabola in standard form.  The "-2" causes the parabola to be steeper and to be concave down since the coefficient is negative.  The "- 1" shifts the parabola to the right 1.  The "+ 5" shifts the parabola up 5.  Thus the vertex is (1,5) and the axis of symmetry is x = 1.

The graph is shown below.

Problem 14

If

f(x)  =  log₃(2x - 1)

find

f ^-1(x)

Solution

First set

y  =  log₃(2x - 1)

And switch the x and y.

x  =  log₃(2y - 1)

Next, since the log is the inverse of the exponential, we can put this into exponential form.

3^x  =  2y - 1

Now add 1 to both sides to get

3^x + 1  =  2y

Finally, divide both sides by 2 to get

Solution 

Since every horizontal line passes through this graph at most once, it passes the horizontal line test.  Thus the function is 1-1.