In the following pages I will work with the randomly selected linear equations: f(x) = x+2 and g(x) = 3x-7. I will explore and observe what types of graphs are obtained for h(x) when we allow h(x) to be:
As we see, h(x) is the sum of the two linear functions: f(x) and g(x). As you will further notice, h(x) is also linear. This will always be true. The sum of two linear functions yields a linear function. Why? Let's look at the general case:
By the property of commutativity, we can rearrange to obtain:
Therefore, h(x) is also a linear function
whose slope is equivalent to the sum of the slope of f(x) plus
the slope of g(x). Just as well, the y-intercept of h(x) is equivalent
to the sum of the y-intercepts of f(x) and g(x).
As you can see the product of the two linear functions (f(x) and g(x)), yields a quadratic function, h(x). Will this always happen? Well, let's look at the general case and see: Suppose f(x) = ax+b and g(x) = cx+d, where a,b,c and d are real numbers. Let h(x) = f(x)g(x)
Well, addition and multiplication of real numbers yield real numbers. Therefore, let
such that p, q and r are real numbers.
Then we have: h(x) = p(x^2) + qx +r. This is the general form
of a quadratic equation.
We can see from our graph that h(x) has an asymptote. But what is the exact value of x where this asymptote occurs? Well, from our knowledge of functions, we know that h(x) is discontinuous when its denominator is equal to zero. This is because division by zero is undefined.
Besides looking at the graph above, how do we know x = 2.333, the point where our denominator is equal to zero, yields an asymptote rather than a whole in the graph or some other form of discontinuity? Let's make a table of the behavior of the function as x approaches 2.333. . . in order to investigate.
As you can see, as x approaches 2.333. . .from the left, h(x) approaches negative infinity (i.e. decreases without bound). Similarly, as x approaches 2.333. . .from the right, h(x) approaches positive infinity (i.e. increases without bound).
Let's look at the general case to see if we can predict where the discontinuity will be for any given function:
Therefore, if h(x) = ax +b/cx+d where f(x) and g(x) are two linear functions, h(x) will always have an asymptote. This asymptote will always be located at x = - d/c.
Simplifying h(x) further, we see that h(x) = 3x -7+2 = 3x -5 is another linear function. Will this always be? Again, in order to decide, let us return to the general case. Suppose h(x) is the composition of the two linear functions: f(x) = ax+b and g(x) = cx+d.
Well, multiplication and addition of real numbers yield real numbers. Therefore, let:
Therefore, we get: h(x) = sx + t, the
general form of a linear function. Therefore, the composition
of two linear functions always yields a third linear function.