Question

In Exercises \(11-18,\) use the function \(f\) defined and graphed below to answer the questions. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {-1 \leq x<0} \\\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x < 3}\end{array}\right.$$ What new value should be assigned to \(f(1)\) to make the new function continuous at \(x=1 ?\)

Short answer

To make the function \(f(x)\) continuous at \(x=1\), assign \(f(1) = 2\).

Step-by-step solution

Understand Continuity

A function \(f(x)\) is said to be continuous at a certain value \(x=a\) if the following three conditions are met: 1. \(f(a)\) is defined. 2. The limit as \(x\) approaches \(a\) exists.3. The limit as \(x\) approaches \(a\) equals \(f(a)\). So to make a function continuous at a certain point, the value of the function at that point should be the same as the limit of the function at that point.

Calculate limit at \(x=1\)

From the right side, i.e, for \(1 < x < 2\), the function is given by \(-2x + 4\). Therefore, the limit of \(f(x)\) as \(x\) approaches \(1\) from the right (denoted by \(\lim_{{x \to 1^+}} f(x)\)) equals to \(-2(1) + 4 = 2\).\nFrom the left side, i.e, for \(0 < x < 1\), the function is \(2x\). So, the limit as \(x\) approaches \(1\) from the left (denoted by \(\lim_{{x \to 1^-}} f(x)\)) equals to \(2(1) = 2\).\nSince, the limit at \(x = 1\) from both sides exists and is the same, we say that \(\lim_{{x \to 1}} f(x) = 2\).

Assign new value to \(f(1)\) to make function continuous

According to the third criteria of continuity, to make the function \(f(x)\) continuous at \(x=1\), \(f(1)\) should be equal to the limit of \(f(x)\) at \(x=1\), so, to make the function continuous at \(x=1\), the new value that should be assigned to \(f(1)\) is \(\lim_{{x \to 1}} f(x) = 2\).