Determine if Continuous f(x)=(2x+1)/(x^2+x-2)

Solution:

Step:1

Identify the domain to check the continuity of the function.

Step:1.1

Equating the denominator of $\frac{2x + 1}{x^2 + x - 2}$ to zero to find discontinuities.

$x^2 + x - 2 = 0$

Step:1.2

Proceed to find the roots of $x$.

Step:1.2.1

Apply the AC method to factor $x^2 + x - 2$.

Step:1.2.1.1

Look for two integers whose product is $c$ and sum is $b$, for the quadratic $x^2 + bx + c$. Here, we need integers with a product of $-2$ and a sum of $1$.

$-1, 2$

Step:1.2.1.2

Express the quadratic in its factored form using the identified integers.

$(x - 1)(x + 2) = 0$ $(x - 1)(x + 2) = 0$

Step:1.2.2

Recognize that if any factor equals zero, the whole expression equals zero.

$x - 1 = 0$ $x + 2 = 0$

Step:1.2.3

Isolate $x - 1$ and solve for $x$.

Step:1.2.3.1

Set $x - 1$ to zero.

$x - 1 = 0$

Step:1.2.3.2

Add $1$ to both sides to solve for $x$.

$x = 1$ $x = 1$

Step:1.2.4

Isolate $x + 2$ and solve for $x$.

Step:1.2.4.1

Set $x + 2$ to zero.

$x + 2 = 0$

Step:1.2.4.2

Subtract $2$ from both sides to solve for $x$.

$x = -2$ $x = -2$

Step:1.2.5

Combine the solutions that satisfy $(x - 1)(x + 2) = 0$.

$x = 1, -2$ $x = 1, -2$

Step:1.3

The domain includes all $x$ values that do not make the denominator zero.

Interval Notation: $(-\infty, -2) \cup (-2, 1) \cup (1, \infty)$ Set-Builder Notation: $\{x | x \neq 1, -2\}$ Interval Notation: $(-\infty, -2) \cup (-2, 1) \cup (1, \infty)$ Set-Builder Notation: $\{x | x \neq 1, -2\}$

Step:2

Given the domain excludes certain real numbers, $\frac{2x + 1}{x^2 + x - 2}$ is not continuous for all real numbers.

Step:3

Knowledge Notes:

To determine the continuity of a function like $f(x) = \frac{2x + 1}{x^2 + x - 2}$, we need to establish its domain, which is the set of all input values $x$ for which the function is defined. A function is continuous if it is defined and smooth (without breaks, holes, or jumps) over its entire domain.

1. Domain of a Function: The set of all possible input values (usually $x$ values) for which the function is defined. For rational functions, the domain is all real numbers except where the denominator is zero.

2. Continuity: A function is continuous at a point if the limit as $x$ approaches the point from both sides is equal to the function's value at that point. A function is continuous on an interval if it is continuous at every point in that interval.

3. Factoring Quadratics: A common method to solve quadratic equations of the form $ax^2 + bx + c = 0$ is to factor the quadratic into a product of binomials. The AC method is a factoring technique used when $a \neq 1$. It involves finding two numbers that multiply to $ac$ and add to $b$.

4. Interval Notation: A way to describe the domain or range of a function using intervals. For example, $(a, b)$ denotes all numbers between $a$ and $b$, not including $a$ and $b$ themselves.

5. Set-Builder Notation: Another way to describe the domain or range of a function. It specifies a property that members of the set must satisfy. For example, $\{x | x \neq a\}$ denotes all numbers except $a$.

In the context of this problem, we are looking for values of $x$ that make the denominator zero, as these are the points where the function is not defined and thus not continuous. Once we find these values, we exclude them from the domain. If the domain is not all real numbers, the function is not continuous over all real numbers.