MATH 130 Fundamentals of Modern Mathematics 1 Assignment 1 2026 | University of Otago

MATH 130 Fundamentals of Modern Mathematics 1 

Question 6

Show that the line determined by the expression

−4x + 2y = 4

is the same as the line

{v + tw : t ∈ R}

where v and w are the vectors v = [1
                                  4] and w = [−3
                                               −6].

[This is one example. Different students will have different lines and vectors.]

Solution:

One way to show that two lines are identical is to pick two distinct points on one line and verify that they also lie on the other line. (Two points determine a line uniquely.)

We can easily obtain two points on the second line by choosing two different values for the parameter t. For example, for t = 0 and t = 1, we get the points

[1
 4] and [1
          4] + [−3
                −6] = [−2
                       −2].

To show that the first point is on the first line as well, we plug in x = 1, y = 4 and check that the equation --4x+2y = 4 is satisfied:

−4x + 2y = −4 + 2 · 4 = −4 + 8 = 4.

In the same way, we get for the second point

−4x + 2y = −4 · (−2) + 2 · (−2) = 8 − 4 = 4.

Since both points satisfy the equation of the first line, the two lines are identical.

Alternative solution: We can also show that every point on the second line satisfies the first equation. A point on the second line has the x and y coordinates x = 1 − 3t, y = 4 − 6t. Plugging this into the first equation, we get

−4x + 2y = −4(1 − 3t) + 2(4 − 6t) = −4 + 12t + 8 − 12t = 4,

i.e. the equation is satisfied independently of the value of t. Hence, both lines coincide.

Second alternative: Using the second line equation, obtain two points. Use these to construct the slope of the line in the form y = mx + c, m = (y₂ − y₁) / (x₂ − x₁) = 2. Plug in one point to obtain c = 2. Hence y = 2x + 2. Rearranging gives the first line equation.

Question 11

A Technique I was taught for converting from Celsius to Fahrenheit is 

Add forty, times by nine over five, and then subtract forty

To convert the other way:

Add forty, times by five over nine, and subtract forty

Express both conversions in terms of functions and use Wikipedia to check if they are correct. Determine whether one function is the inverse of the other function. For all parts of this question, explain and justify your answer carefully and clearly.

Solution:

For converting from Celsius to Fahrenheit, we define a function y = f(x) that takes the temperature in Celsius x and returns the temperature in Fahrenheit y. Following the instructions, we start from the Celsius value x and add 40, giving x + 40. Then we multiply by 9/5, so we obtain 9/5(x + 40). Finally, we subtract 40, giving 9/5(x + 40) − 40. Altogether, we have

f(x) = 9/5 (x + 40) − 40,

which simplifies to

f(x) = 9/5 x + 32.

Similarly, for the conversion from Fahrenheit to Celsius, we define a function y = g(x). This time, x is the temperature in Fahrenheit and y the temperature in Celsius. Starting from x, we add 40, giving x + 40. We multiply by 5/9 to get 5/9(x + 40). Then we subtract 40, giving 5/9(x + 40) − 40, so

g(x) = 5/9 (x + 40) − 40 = 5/9 (x − 32).

On Wikipedia, we find:

[°C] = ([°F] − 32) × 5/9,   [°F] = [°C] × 9/5 + 32.

The first formula corresponds to the function g, and the second to the function f (in slightly different notation).

To see if g is the inverse function of f, i.e. g = f⁻¹, we need to check if (g ∘ f)(x) = x for all x. We have

(g ∘ f)(x) = 5/9 ([9/5 x + 32] − 32) = 5/9 (9/5 x) = x

as required, so g is the inverse of f (and f is the inverse of g).

(Alternatively, check that f is the inverse of g by showing (f ∘ g)(x) = x, or construct the inverse of f(x) by solving y = 9/5 x + 32 for x and swapping x and y. The result will be g(x).)