An equation expresses the relation between variables. A relation between y and x can be drawn as a graph in the xy plane.
In order to do that, you want to "plug in" a value for x, and then get the result as a value for y. With an equation that expresses y in terms of x, you can do just that. For example, plug in the value x=3 in the equation y=2x+4 easily gives you y=10. You plot the point (3,10) and repeat for different x.
If y is not clearly expressed in terms of x, you get equations like: 2x + 4y + x/y = 0. You cannot plug in an x and calculate a y in an easy way.
Most of the time you'll be able to rewrite the equation in the right "pluggable" form. In fact, when you have an equation in this form, it maps one to one to a function f(x). y=2x+4 and f(x)=2x+4 is kind of the same thing. "f is a function of x" and "y is expressed in terms of x" are similar statements.
Answer:
1. f(x) is continuous at x = 1
2. f(x) is continuous at x = 1
Step-by-step explanation:
Determine whether the function is continuous or discontinuous at x=1.
Examine the three conditions in the definition of continuity.
1. f(x) = x²+8 if x<1
2. f(x) = 6x² - 3 if x> 1
For a function to be continuous at a given x-value
then lim x→a f(x) = f(a)
Meaning that
What this is saying is that, as x gets closer to a , f(x) should also get closer to f(a).
1. Limit of f(x) = x² + 8 at x = 1
= 1²+8 = 9
f(a) = f(1) = 1² + 8 = 9.
lim x→a f(x) = f(a) = 9
2. Limit of f(x) = 6x² - 3 at x = 1
= 6(1²) - 3 = 3
f(a) = f(1) = 6(1²) - 3 = 3
lim x→a f(x) = f(a) = 3
Add three to both sides of the equation
