Find the constant a such that the function f(x) = -12...

Find the constant $a$ such that the function \begin{equation*} f(x) = \begin{cases} -12 \cdot \frac{\sin x}{x}, & x < 0 \\ a + 12x, & x \ge 0 \end{cases} \end{equation*} is continuous on the entire real line.

Transcript

00:01 We are going to find the constant a such that the function is continuous on the entire real line and the function is defined as 8 sin x divided by x if x is negative and a minus 9x if x is greater than or equal to 0.

00:22 So by this definition the function is defined on the entire real line so the domain of f is the real numbers.

00:30 The image of 0 is the constant a and then we want to see which value of a we get to impose in such a way that this function will be continuous in the entire real line.

00:48 So first of all we note that for any x different from 0 at any number different from 0 the function is continuous because this formula here 8 sin x over x is a continuous expression for any x different from 0 and the formula down here is a polynomial of degree 1 so it is obviously continuous on the real numbers in particular any number different from 0.

01:26 So this function is clearly continuous at every value of x different from 0.

01:34 It is clear that f is continuous at every x different from 0.

02:01 Saying another way or putting it in another way this function can be discontinuous maybe at 0 that is the only possibility for discontinuity of the function simply because it is changing definition at that point and the formulas defining the function in other points different from 0 are formulas that are continuous functions.

02:39 So f could only be discontinuous at x equals 0.

02:56 One way it could be discontinuous at 0 is that it is not defined at 0.

03:01 It gets to be defined at 0 to be continuous at 0 but it is because the domain we saw is the real numbers...

Question

Find the constant $a$ such that the function \begin{equation} f(x) = \begin{cases} -12 \cdot \frac{\sin x}{x}, & x < 0 \\ a + 12x, & x \ge 0 \end{cases} \end{equation} is continuous on the entire real line.

Question

Find the constant $a$ such that the function \begin{equation*} f(x) = \begin{cases} -12 \cdot \frac{\sin x}{x}, & x < 0 \\ a + 12x, & x \ge 0 \end{cases} \end{equation*} is continuous on the entire real line.

Please give Ace some feedback

Find the constant $a$ such that the function \begin{equation} f(x) = \begin{cases} -12 \cdot \frac{\sin x}{x}, & x < 0 \\ a + 12x, & x \ge 0 \end{cases} \end{equation} is continuous on the entire real line.