# Math Homework 1 Solutions

1. Find:

Solution |

2. Plot the following pairs of points and find the corresponding line equations:

(a) (3,6); (7,14)

(b) (-1,3); (4,-10)

Solution |

The following plots shows the points and the corresponding line for (a) and (b).

(a) We will use the equation for a line, y = mx + b. First we find the slope, m.

We plug m = 2 into the equation y = mx + b and get y = 2 ·
x + b = 2x + b.

To find b we will plug in x = 3 and y = 6 from the point (3,6) and solve for b.

y = 2x + b

6 = 2 · 3 + b

6 = 6 + b (subtract 6 from each side)

0 = b.

Plugging b = 0 into y = 2x + b we get that the equation of the line is y = 2x + 0, or y = 2x.

(b) We will use the equation for a line, y = mx + b. First we find the slope, m.

We plug into the
equation y = mx + b and get .

To find b we will plug in x = -1 and y = 3 from the point (-1,3) and solve for
b.

(subtract
from each side)

Plugging into
we get that the equation of the line is
, or

y = -2.6x + .4.

3. Solve the following equations for their roots:

Solution |

(a) To solve this, we want to get x by itself.

4x + 8 = 0 (subtract 8 from each side)

4x = -8 (divide each side by 4)

x = -2

(b) To solve this, we want to get x by itself.

-3x + 1 = 5 (subtract 1 from each side)

-3x = 4 (divide each side by -3)

(c) To solve this we will use the quadratic formula. If
our equation is in the from ax^{2} + bx + c = 0,

then our solution(s) are

In this case a = 1, b = 4, and c = -5.

So x = -5 and x = 1.

(d) Again we will use the quadratic formula. Notice that
we want our equation in the form

ax^{2} +bx+ c = 0, so we first subtract 3 from both sides of the
equation to get that we are solving

Plugging into the quadratic formula with a = 3, b = 5, and c = -11, we get

So and
.

4. Find the Domain and Range for each function (Graph first);

Evaluate each function (if you can) at x = -2,1,5.

Solution |

The following graphs show the functions for (a), (b), and (c).

(a) The domain of this function is (-∞,∞) (all real
numbers). The range is also [-∞,∞). We can

evaluate at x = -2, 1, 5 by plugging in for x.

(b) The domain of this function is (-∞,-2) and (-2,∞) (all
real numbers except -2). The range

is (-∞, 0) and (0,∞) (all real numbers except 0). We can evaluate at x = 1, 5 (x
= -2 is not in

the domain) by plugging in for x.

(c) The domain of this function is (-∞, 5) (all real
numbers less than 5). The range is (-∞,∞) (all

real numbers). We can evaluate at x = -2, 1 (x = 5 is not in the domain) by
plugging in for x.

5. For each f(x), write down a simpler function and how it was transformed. Graph f(x).

Solution |

Graphs of the functions are shown below. In each case, the
function f(x) is the solid

line, while the simpler function g(x) is the dashed line.

(a) A simpler function is g(x) = x^{2}. The
function f(x) is the function g(x) shifted right by 1 unit and

down by 2 units.

(b) A simpler function is g(x) = log(x). The function f(x) is the function g(x)
shifted left by 4 units

and stretched vertically by a factor of 2.

(c) A simpler function is g(x) = e^{x}. The function f(x) is the
function g(x) shifted right by 1 unit, up by

3 units, and then reflected about the x-axis.

6. Find the limit of each function:

Solution |

In each case, we will be looking at plots of the function. These are shown below.

(a) Look at a plot of f(x) = x^{2} + 4x + 3 at x =
2 (see plot (a) above). The plot is continuous and finite

at x = 2, so the limit is simply f(2). That is,

(b) Look at a plot of f(x) = log x at x = 0 (see plot (b)
above). From the plot, it looks like as x is

getting closer to 0, the function is going off to -∞. So our limit is

(c) Look at a plot of f(x) = e^{-x} as x → ∞
(see plot (c) above). That is, examine the plot as x gets

really large. From the plot it look like as x is getting large, the function is
getting closer and closer

to zero. So our limit is

(d) Look at a plot of
as x 3 (see plot (d) above). That is,
examine the plot as x gets close

to 3 coming from the right-hand side. From the plot it look like as x gets close
to 3 from the right,

the function is getting larger and larger. So our limit is

7. A population starts at 50 at time t = 0. The suggested
population model is , where t

is in years.

(a) What will the population be in 2 years?

(b) How long will it take for the population to reach 125?

(c) What is the limit of P(t) as t → ∞?

Solution |

(a) Here we just want to evaluate P(2).

individuals.

(b) Now we want to find t such that P(t) = 125. So we solve for t.

(multiply each side by 1 + 3e^{-t})

(subtract
125 from each side)

(divide
each side by 375)

(take the
natural log of each side)

-t = -1.609 (divide each side by -1)

(c) We already know from problem 5(c) that e^{-t} →
0 as t → ∞. So as t → ∞, we have

So the limit of the population is 200 individuals.