Investigation with linear functions

By: Lauren Lee

Make up linear functions f(x) and g(x). Explore, with different pairs of f(x) and g(x) the graphs for

i. h(x) = f(x) + g(x)

ii. h(x) = f(x).g(x)
iii. h(x) = f(x)/g(x)
iv. h(x) = f(g(x))

Example 1. Let f(x) = 2x + 1 and g(x) = 3x + 4. Graph the sum, product, quotient of f and g, and composite f(g(x)).

a. f(x) + g(x) = (2x +1) + (3x + 4)

b. f(x)g(x) = (2x+1)(3x +4)

c. f(x)/g(x) = (2x+1)/(3x +4)

d. f(g(x)) = 2(3x +4)+1

Let’s look at the graphs!

Example 2. Let f(x) = 3x - 2 and g(x) = x + 1. Graph the sum, product, quotient of f and g, and composite f(g(x)).

a. f(x) + g(x) = (3x -2) + (x +1)

b. f(x)g(x) = (3x -2)(x +1)

c. f(x)/g(x) = (3x -2)/(x +1)

d. f(g(x)) = 3(x +1) –2

Look at the graphs!

Let’s try one more example! Let f(x) = 2x – 4 and g(x) = x – 1

a. f(x) + g(x) = (2x -4) + (x -1)

b. f(x)g(x) = (2x -4)(x -1)

c. f(x)/g(x) = (2x -4)/(x -1)

d. f(g(x)) = 2(x -1) –4

Let’s look at the graphs again!

So what can we conclude about compositions of linear functions from looking at the graphs?

First of all, we know that the sum of two linear functions is a linear function (y = ax + b). Look at example 1 again: f(x) + g(x) = (2x +1) + (3x +4) = 5x + 5, which is in the form of y = ax + b. Look at part a. of the graphs to see this also!

We also know that the product of two linear functions is a quadratic function. If you look at part b. in the graphs, you will notice that they are all parabolas. From example 1 again, f(x)g(x) = (2x +1)(3x +4) = 6x^2 + 11x + 4.

What can we say about the quotient of two linear functions? Look at the asymptotes of part c. in the graphs. The vertical asymptotes of f(x)/g(x) are the real roots of g(x).

And finally, the composite function of two linear functions is linear. Look at part d. of the graphs to see this!

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