Question

What would a graph look like for this function? Also, is it continuous?

Answer 1

Answer: The function is continuous.

The graph is below.

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Step-by-step explanation:

The piecewise function is defined like so

if x < -4, then f(x) = 6. This is the same as saying y = 6 or y = 0x+6. This graphs out a straight horizontal line through 6.

If
`-4 \le x < 4`, then f(x) = -10+x^2. This produces the red portion of the graph which is the parabolic piece. It's only defined when x is between -4 and 4. We include x = -4 but exclude 4.

The last piece is f(x) = 2x-2 and that occurs when
`x \ge 4` and this is the blue piece of the graph shown below.

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The last section talks about the graph. Let's now go over if this function is continuous or not.

Each piece mentioned is a polynomial. All polynomials are continuous. This means that that any given piece itself on its own is something we don't need to worry about. We simply need to worry if the pieces connect or not.

If x = -4, then plugging it into the second piece leads us to...

f(x) = -10+x^2

f(-4) = -10+(-4)^2

f(-4) = 6

This output 6 matches perfectly with the '6' given at the first piece of the piecewise function. Therefore, the first and second pieces connect together.

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Like the previous section, we need to check if the second and third pieces connect or not.

Plug x = 4 into the second piece

f(x) = -10+x^2

f(4) = -10+(4)^2

f(4) = 6

Repeat for the third piece

f(x) = 2x-2

f(4) = 2(4)-2

f(4) = 6

Both outputs match up, so these pieces connect as well.

The graph shows this that the three pieces connect to form one continuous curve of sorts. There are no gaps, jumps, holes, etc to indicate discontinuity.

Therefore, this function is fully continuous everywhere.