Let f(x) = [2x² + 1] and g(x) = { 2x - 3, x

Q: Wait, why do we check $x=0$ separately? Can't we just find the range for the whole interval $(-1, 1)$?

Great question! The function $g(x)$ is defined differently for $x < 0$ and $x \ge 0$. Because the definition of the inner function changes at $x=0$, the behavior of the composite function $f(g(x))$ could potentially change there too. We must verify if the limit from the left equals the limit from the right at this transition point to ensure we don't miss a discontinuity or incorrectly count one.

Q: Does the greatest integer function $[u]$ jump at every integer $u$?

Yes, exactly! The function $f(u) = [u]$ is discontinuous at every integer value of $u$. In our composite function $f(g(x))$, the discontinuities occur whenever the inner expression $2(g(x))^2 + 1$ hits an integer. By finding the range of this inner expression, we identify all the integers it passes through, and each one corresponds to a point of discontinuity.

Q: Why did the range of $g(x)^2$ stay the same for both branches even though $g(x)$ was different?

It is a beautiful coincidence of the problem's design! On the left branch, $g(x)$ maps to $(-5, -3)$, and on the right, it maps to $(3, 5)$. When you square these values, both intervals map to $(9, 25)$. This symmetry is why both branches contribute the same number of discontinuities. Always look for such patterns; they make the math much more elegant!

Question

Let $f(x) = [2x^2 + 1]$ and $g(x) = \begin{cases} 2x - 3, & x < 0 \ 2x + 3, & x \ge 0 \end{cases}$, where $[t]$ is the greatest integer $\le t$. Then, in the open interval $(-1, 1)$, the number of points where $fog$ is discontinuous is equal to ______.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The function under investigation is the composition $f(g(x))$, where $f(x) = [2x^2 + 1]$. We are analyzing the discontinuities of this composite function over the interval $x \in (-1, 1)$.

The greatest integer function $[u]$ is discontinuous whenever $u$ is an integer. Therefore, the composite function $f(g(x))$ is discontinuous whenever $2(g(x))^2 + 1 = k$, where $k$ is an integer.

### The Left Branch: The Descent into the Negative

For $x \in (-1, 0)$, the inner function is defined as $g(x) = 2x - 3$. As $x$ increases from $-1$ to $0$, $g(x)$ increases from $-5$ to $-3$.

The range of $g(x)$ is the interval $(-5, -3)$. Consequently, the range of $(g(x))^2$ is $(9, 25)$.

Applying the transformation $2(g(x))^2 + 1$, we find the range of the inner expression:
$$2(9) + 1 < 2(g(x))^2 + 1 < 2(25) + 1$$
$$19 < 2(g(x))^2 + 1 < 51$$

Since the expression $2(g(x))^2 + 1$ is strictly monotonic on this interval, it takes on every integer value $k \in \{20, 21, \dots, 50\}$ exactly once. This yields $50 - 20 + 1 = 31$ points of discontinuity.

### The Right Branch: The Ascent into the Positive

For $x \in (0, 1)$, the inner function is defined as $g(x) = 2x + 3$. As $x$ increases from $0$ to $1$, $g(x)$ increases from $3$ to $5$.

The range of $g(x)$ is the interval $(3, 5)$. Squaring this gives $(g(x))^2 \in (9, 25)$.

Following the same algebraic transformation, the inner expression $2(g(x))^2 + 1$ spans the interval $(19, 51)$. Similar to the left branch, the function crosses every integer $k \in \{20, 21, \dots, 50\}$ exactly once. This contributes another $31$ points of discontinuity.

### The Bridge: The Critical Point at $x=0$

We must verify the continuity at the junction $x=0$. We evaluate the function value and the limits:

1. **Function Value:** $g(0) = 3$, so $f(g(0)) = [2(3)^2 + 1] = [19] = 19$.
2. **Left-hand Limit:** As $x 	o 0^-$, $g(x) 	o -3^+$, so $(g(x))^2 	o 9^+$. Thus, $2(g(x))^2 + 1 	o 19^+$, and $[19^+] = 19$.
3. **Right-hand Limit:** As $x 	o 0^+$, $g(x) 	o 3^+$, so $(g(x))^2 	o 9^+$. Thus, $2(g(x))^2 + 1 	o 19^+$, and $[19^+] = 19$.

Since the left-hand limit, right-hand limit, and function value are equal, the function is continuous at $x=0$.

### The Final Tally

Summing the discontinuities found in each branch:
$$31 	ext{ (left)} + 31 	ext{ (right)} + 0 	ext{ (at } x=0) = 62$$

The total number of points of discontinuity for the composite function on the interval $(-1, 1)$ is **62**.

Enter Numerical Value:

Visualized Solution

Defining the Composite Function $f (g (x))$

Analyzing the Left Branch: $x \in (- 1, 0)$

Range of the Inner Expression for $x < 0$

Discontinuities in $(- 1, 0)$

Analyzing the Right Branch: $x \in (0, 1)$

Range of the Inner Expression for $x > 0$

Discontinuities in $(0, 1)$

Checking Continuity at $x = 0$

Final Calculation

The Sigma Insight: Continuity at a Point and in an Interval

Analyzing the Setup

The Left Branch

The Right Branch

The Bridge

The Final Tally

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Enter Numerical Value:

Visualized Solution

Defining the Composite Function .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(g(x))

Analyzing the Left Branch: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x∈(−1,0)

Range of the Inner Expression for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x<0

Discontinuities in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (−1,0)

Analyzing the Right Branch: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x∈(0,1)

Range of the Inner Expression for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x>0

Discontinuities in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (0,1)

Checking Continuity at .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=0