Question

The function f is defined as follows
f(x) { 3+x if x< 0
{x^2 if x > 0
(a) Find the domain of the function
(b) Locate any intercepts
(c) Graph the functions
(d) Based on the graph, find the range
(e) Is f continuous on its domain?

Answer 1

ok, so is a "piecewise" function, it has 2 patterns
so let's find the domain

for
f(x) { 3+x if x< 0
{\(x^2\) if x > 0

what values can "x" take based on those "if" restrictions?

Answer 2

I have no idea is it 0

Answer 3

well, can we give x = 25?

Answer 4

is there anything there saying we can't?

Answer 5

no

Answer 6

or say let's make x = 15, or 18, or -25, or -15, or -18
is there anything saying we can't?

Answer 7

no ma'am

Answer 8

the function
f(x)
{ 3+x if x< 0 # x is less than 0, not equal to 0, just less, how less? up to \(-\infty\) it seems

{x2 if x > 0 # x is greater than 0, not equal to 0, just greater, how much bigger? up to \(+\infty\) it seems

Answer 9

so, x can go to infinity on either side, BUT never be 0
so, that's the DOMAIN

Answer 10

or in interval notation will be \((-\infty, 0)\) and \((0, +\infty)\)

Answer 11

open intervals, because, it never gets to 0, and infinity is really never reachable either

Answer 12

okay @ jdoe, the answer is (00,0) and (0,00)? I know what the double o's, cant implement it on the keyboard

Answer 13

see above :), yes

Answer 14

The domain can be expressed as {x | x < 0} u {x|x>0} too

Answer 15

okay thanks guys!!!

Answer 16

so, since x will never become 0, that means, you won't have a y-intercept, because y-intercepts occur only when x = 0, and in this case, x cannot be =0

what about x intercepts?
x intercepts occur when y is set to 0
so let's check the 1st one

y = 3+x
0 = 3+x => x = -3 # so that's one x intercept, at (-3, 0)

let's check the 2nd one
y = \(x^2\)
0 = \(x^2 \implies x =0\) # no dice because x cannot be =0, we're restricted there
# so no x intercepts on the 2nd one, just the 1st one

Answer 17

okay.... got it

Answer 18

now you'd just need to graph them, keeping in mind their range
so it looks more or less like

Answer 19

Wow thanks, you learn something new everyday...:)

Answer 20

is a "piecewise" function, so it has 2 patterns, one for x<0, and another pattern for x>0

is really an inclined line when x<0
and is a parabola when x> 0

Answer 21

okay....

Answer 22

based on the graph, let's find the range
what values is "y" taking?
the inclined line on the left is going and going down and down further down to \(-\infty\)

the parabola on the right, is going up and up and up and up further up to \(+\infty\)

Answer 23

and that's the range, "y" is really "occupying" from \((-\infty, +\infty)\)

Answer 24

now, if the piecewise function "continuous"
as in in a single continum, as in an unbroken thread
well, just look at the graph, that'd tell you :)

Answer 25

okay thanks jdoe your the best!!!

Answer 26

yw

Answer 27

@charisse1 if you ever see something that you don't know how to make, select it, then right-click and choose Show Math As>TeX Commands and you'll get a little pop-up menu that tells you what you need to type in LaTeX to generate it. For example,

\[(-\infty, +\infty)\]is created with
(-\infty, +\infty)

Answer 28

uh, not a pop-up menu, a pop-up window...