Let’s examine what happens when we combine two linear functions using basic computations.
We want to explore these 4 different cases
l. f(x) + g(x)
ii. f(x) * g(x)
Let f(x)= x+1 and g(x)= 3x-4
i. What happens when you add two linear functions? What will the graph of the result look like? Is it linear? Explain why or why not.
The red line represents f(x) and the blue line represents g(x). The purple is represents the function created by adding f(x) and g(x). It is still linear because when you add variables you only add coefficients. The power on the variable will stay the same.
ii. What happens when you multiply two linear functions? Can you use the laws of exponents to predict what your graph will look like?
When you multiply two linear functions you will create a quadratic function. The law of exponents states that when you multiply variables, you add the powers. Therefore the power on the x becomes 2.
iii. Now try dividing the functions.
This graph is called a hyperbola. There is an interesting gray line on the graph. Where does it occur (write the equation for this gray line). What significance does this line have? Why is at this specific point? How do we find where the graph will cross the x and y axis? How can you see these points in the equation?
The equation for the gray line is x=. This is the value for x where the denominator will equal zero. The function is undefined at this point. Therefore the red lines will not cross this vertical line. This is called the vertical asymptote. There is also a horizontal asymptote. In order to find the horizontal asymptote, you must first solve for x and then find where the function will be undefined (the value of y that will make the denominator 0).
Y= Therefore the function will
Y(3x-4)=1/2x +1 be undefined when
The place where the graph crosses the x- and y-axis are called the x-intercept and the y-intercept. To find the x-intercept you allow y to equal 0. In order to proceed, you will be multiplying the denominator by 0 so you can just set the numerator equal to 0 to solve for the x-intercept. Conversely allow x to equal 0 in order to find the y-intercept. For this equation h(x)=the x-intercept is (-2, 0) and the y-intercept is (0, -1/4). You can see these points easily in the equation by setting the numerator equal to 0 (this will produce the y-intercept) and by eliminating the expressions with the variable x (which will provide the x-intercept).
iv. What does the graph of f(g(x)) look like? Is it linear? Why? How do we find this function algebraically?
In order to find this function algebraically, you must replace the x in the f(x) function with the equation for g(x). So f(g(x))=1/2(3x-4)+1 and once simplified it becomes (f(g(x))=3/2x-1. The new equation is linear because you are merely replacing one x with another value of x. The two x’s are not being multiplied together so the power remains 1.