Math worksheet focusing on identifying intervals of continuity and classifying discontinuities for various functions.

Let’s go through each function one by one to find where it is continuous and classify any discontinuities.

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1. r(x) = (4x|x + 2|) / (x + 7)

- The denominator is zero when x = -7 → so the function is undefined there.
- Absolute value |x+2| is always defined, no problem.
- So only issue is at x = -7.
- As x approaches -7 from left or right, numerator approaches 4*(-7)*|-5| = -28*5 = -140 ≠ 0.
- Denominator approaches 0 → so vertical asymptote → essential discontinuity.
- Continuous everywhere else: (-∞, -7) ∪ (-7, ∞)

✔ Final for r(x):
Intervals: (-∞, -7), (-7, ∞)
Discontinuity at x = -7 → essential

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2. n(x) = { x² -13x -32 if x ≤ -8 ; x + 144 if x > -8 }

Check continuity at x = -8 (only possible break point).

Left side (x ≤ -8): plug in x = -8 → (-8)² -13*(-8) -32 = 64 + 104 -32 = 136
Right side (x > -8): limit as x→-8⁺ of (x + 144) = -8 + 144 = 136
Function value at x=-8 is 136 (since defined by first piece)

So left limit = right limit = function value → continuous at x = -8

No other breaks → continuous on all real numbers.

✔ Final for n(x):
Intervals: (-∞, )
No discontinuities

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3. e(x) = x² / [(x + 2)(x + 4)]

Denominator zero when x = -2 or x = -4 → undefined there.

Numerator at those points:
- At x = -2: (-2)² = 4 ≠ 0 → so vertical asymptotes → essential discontinuities
- At x = -4: (-4)² = 16 ≠ 0 → same → essential

Continuous elsewhere: (-∞, -4) ∪ (-4, -2) ∪ (-2, ∞)

✔ Final for e(x):
Intervals: (-∞, -4), (-4, -2), (-2, ∞)
Discontinuities at x = -4 and x = -2 → both essential

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4. c(x) = { x³ -17x² -26x +31 if x ≤ -7 ; x -956 if x > -7 }

Check at x = -7.

Left side: plug in x = -7 into cubic:

(-7)³ = -343
-17*(49) = -833
-26*(-7) = +182
+31

Total: -343 -833 +182 +31 = let's compute step by step:

-343 -833 = -1176
-1176 +182 = -994
-994 +31 = -963

Right side: limit as x→-7⁺ of (x - 956) = -7 -956 = -963

Function value at x=-7 is -963 (from first piece)

So left limit = right limit = function value → continuous at x = -7

No other issues → continuous everywhere.

✔ Final for c(x):
Intervals: (-∞, ∞)
No discontinuities

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5. u(x) = [(x + 4)(x - 6)] / (x - 7)

Denominator zero at x = 7 → undefined.

Numerator at x=7: (11)(1) = 11 ≠ 0 → so vertical asymptote → essential discontinuity.

Continuous elsewhere: (-∞, 7) ∪ (7, ∞)

✔ Final for u(x):
Intervals: (-∞, 7), (7, ∞)
Discontinuity at x = 7 → essential

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6. s(x) = [5x(x + 1)/|x - 5|] + 2

Denominator |x - 5| = 0 when x = 5 → undefined.

As x → 5, numerator → 5*5*6 = 150 ≠ 0 → so vertical asymptote → essential discontinuity.

Also, absolute value doesn’t cause extra issues — just makes denominator positive except at 5.

Continuous elsewhere: (-∞, 5) ∪ (5, ∞)

✔ Final for s(x):
Intervals: (-∞, 5), (5, ∞)
Discontinuity at x = 5 → essential

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7. g(x) = 3(x² + 81) / |x² - 4|

Denominator |x² - 4| = 0 when x² = 4 → x = ±2

At x = 2: numerator = 3(4 + 81) = 3*85 = 255 ≠ 0 → vertical asymptote → essential
At x = -2: same numerator → 255 ≠ 0 → essential

Absolute value doesn’t remove the zero — still undefined at ±2.

Continuous elsewhere: (-∞, -2) ∪ (-2, 2) ∪ (2, ∞)

✔ Final for g(x):
Intervals: (-∞, -2), (-2, 2), (2, ∞)
Discontinuities at x = -2 and x = 2 → both essential

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8. r(x) = { x³ +54x² -14x -28 if x ≤ 4 ; x² +39x +17 if x > 4 }

Check at x = 4.

Left side: plug in x=4 into cubic:

4³ = 64
54*(16) = 864
-14*4 = -56
-28

Total: 64 + 864 = 928; 928 -56 = 872; 872 -28 = 844

Right side: limit as x→4⁺ of (x² +39x +17) = 16 + 156 + 17 = 189

Wait! 844 ≠ 189 → so left limit ≠ right limit → jump discontinuity

Function value at x=4 is 844 (from first piece), but right side jumps to 189.

So jump discontinuity at x=4.

Continuous elsewhere: (-∞, 4) ∪ (4, ∞)

✔ Final for r(x):
Intervals: (-∞, 4), (4, ∞)
Discontinuity at x = 4 → jump

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Final Answer:

r(x) = (4x|x + 2|)/(x + 7):
Intervals: (-∞, -7), (-7, ∞); Discontinuity at x = -7 → essential

n(x) = piecewise:
Intervals: (-∞, ∞); No discontinuities

e(x) = x²/[(x+2)(x+4)]:
Intervals: (-∞, -4), (-4, -2), (-2, ∞); Discontinuities at x = -4, -2 → essential

c(x) = piecewise:
Intervals: (-∞, ∞); No discontinuities

u(x) = [(x+4)(x-6)]/(x-7):
Intervals: (-∞, 7), (7, ∞); Discontinuity at x = 7 → essential

s(x) = [5x(x+1)/|x-5|] + 2:
Intervals: (-∞, 5), (5, ∞); Discontinuity at x = 5 → essential

g(x) = 3(x²+81)/|x²-4|:
Intervals: (-∞, -2), (-2, 2), (2, ∞); Discontinuities at x = -2, 2 → essential

r(x) = piecewise (last one):
Intervals: (-∞, 4), (4, ∞); Discontinuity at x = 4 → jump