Question Statement

Determine whether the function $f(x)=x^2+1$ is continuous on the following intervals:

(a) $[-1,3]$

(b) $[3,\infty )$

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Background and Explanation

Polynomial functions are continuous everywhere on the real number line $\mathbb{R}$. This fundamental property means that any polynomial is automatically continuous on every possible interval, closed or open, within its domain.

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Solution

Part (a): Interval $[-1,3]$

First, we identify the nature of the function. The function $f(x)=x^2+1$ is a polynomial function (specifically, a quadratic polynomial).

A key theorem in calculus states that all polynomial functions are continuous on the entire real line $(-\infty ,\infty )$. This is because polynomials are built from sums and products of continuous functions (power functions and constants), and combinations of continuous functions remain continuous.

Since $[-1,3]$ is a closed interval contained within $\mathbb{R}$, and $f(x)$ is continuous everywhere on $\mathbb{R}$, it follows immediately that:

$f(x)\text{ is continuous on }[-1,3]$

Part (b): Interval $[3,\infty )$

Similarly, for the interval $[3,\infty )$:

As established in part (a), $f(x)=x^2+1$ is a polynomial function and therefore continuous on all real numbers.

The interval $[3,\infty )$ is a subset of $\mathbb{R}$ (representing all real numbers $x$ such that $x\ge 3$). Since the function is continuous on the entire real line, it must also be continuous when restricted to this half-infinite interval.

Therefore:

$f(x)\text{ is continuous on }[3,\infty )$

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Key Formulas or Methods Used

• Polynomial Continuity Theorem: If $f(x)$ is a polynomial function, then $f$ is continuous on $(-\infty ,\infty )$
• Restriction Property: If a function is continuous on a set $S$, then it is continuous on any subset of $S$
• Interval Notation: Closed intervals $[a,b]$ and half-infinite intervals $[a,\infty )$ are standard domains for continuity analysis

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Summary of Steps

1. Identify the function type: Recognize $f(x)=x^2+1$ as a polynomial (quadratic) function
2. Apply the polynomial continuity theorem: Recall that polynomials are continuous everywhere on $\mathbb{R}$
3. Analyze interval $[-1,3]$: Since this closed interval is contained in $\mathbb{R}$, the function is continuous here
4. Analyze interval $[3,\infty )$: Since this unbounded interval is also contained in $\mathbb{R}$, the function remains continuous here
5. Conclusion: The function is continuous on both specified intervals