Math  /  Algebra

QuestionThe graph of the function ff is shown above. On which of the following intervals is ff continuous? (A) (1,1)(-1,1) (B) (1,2)(1,2) (C) (2,3)(2,3)
D (3,5)(3,5)

Studdy Solution

STEP 1

What is this asking? Which part of this graph is a smooth, unbroken line? Watch out! Remember, a function is continuous if you can draw it without lifting your pen!
Holes, jumps, and asymptotes break continuity.

STEP 2

1. Analyze Interval A
2. Analyze Interval B
3. Analyze Interval C
4. Analyze Interval D

STEP 3

Let's look at the interval (1,1)(-1, 1).
Does the graph of ff have any breaks or jumps in this interval?

STEP 4

Yes! There's a **vertical asymptote** at x=1x = 1.
This means the function explodes towards infinity as xx approaches **one**!
So, ff is *not* continuous on (1,1)(-1, 1).

STEP 5

Now, let's check out the interval (1,2)(1, 2).
Is ff continuous here?

STEP 6

Well, the graph has a **hole** at x=2x = 2.
This is a **point discontinuity**, meaning ff is *not* continuous on (1,2)(1, 2) either.

STEP 7

Time for interval (2,3)(2, 3).
Is ff continuous here?

STEP 8

Nope! There's a **jump discontinuity** between x=2x = 2 and x=3x = 3.
The function jumps from one value to another.
So, ff is *not* continuous on (2,3)(2, 3).

STEP 9

Finally, let's investigate the interval (3,5)(3, 5).
Is ff continuous here?

STEP 10

Yes! Between x=3x = 3 and x=5x = 5, the graph of ff is a smooth, unbroken curve.
There are no holes, jumps, or asymptotes.
So, ff *is* continuous on (3,5)(3, 5).

STEP 11

The function ff is continuous on the interval (3,5)(3, 5), which corresponds to answer choice (D).

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