Prime factorization 64 ladder diagram
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For 3) check the picture, on the right-hand-side
now for 4) check the picture on the left-hand-side
if you notice the intervals, it goes from -∞ and stops at -2, then goes from 2 over to 3, then from 3 onwards, so the domain will also be (-∞ , +∞).
Now, the range...
the 1st subfunction 3-x is just a line coming down form -∞....
the 2nd subfunction is just 2x, also a line going up, now once you get to
the 3rd subfunction is just 5, meaning y = 5, or f(x) = 5, which is just a horizontal line at y = 5.
since the 3rd subfunction goes from x > 3 onwards to +∞, and is just a horizontal line, it doesn't go below 5, so one would think the range is -∞, 5.
however, from the 2nd subfunction if we use say x = -1.9999, 2x ---> 2(-1.9999) = -3.9998 , that's lower than 5, and that's the lowest the piece-wise goes, thus, the range comes from -∞ and goes as low as -3.9998, (-∞ ,-3.9998]
notice, we didn't use -2, because the 2nd subfunction 2x, doesn't include it in its range -2 < x ⩽ 3.
now, for 4) I used -1.9999 but your teacher might be expecting something like -1 or thereabouts, just an integer amount. But the idea being, the 2nd subfunction, does not include the -2.
now for 4) check the picture on the left-hand-side
if you notice the intervals, it goes from -∞ and stops at -2, then goes from 2 over to 3, then from 3 onwards, so the domain will also be (-∞ , +∞).
Now, the range...
the 1st subfunction 3-x is just a line coming down form -∞....
the 2nd subfunction is just 2x, also a line going up, now once you get to
the 3rd subfunction is just 5, meaning y = 5, or f(x) = 5, which is just a horizontal line at y = 5.
since the 3rd subfunction goes from x > 3 onwards to +∞, and is just a horizontal line, it doesn't go below 5, so one would think the range is -∞, 5.
however, from the 2nd subfunction if we use say x = -1.9999, 2x ---> 2(-1.9999) = -3.9998 , that's lower than 5, and that's the lowest the piece-wise goes, thus, the range comes from -∞ and goes as low as -3.9998, (-∞ ,-3.9998]
notice, we didn't use -2, because the 2nd subfunction 2x, doesn't include it in its range -2 < x ⩽ 3.
now, for 4) I used -1.9999 but your teacher might be expecting something like -1 or thereabouts, just an integer amount. But the idea being, the 2nd subfunction, does not include the -2.