Problem 51
Question
Find the values of \(x\) for which each function is continuous. \(f(x)=\frac{2 x+1}{x^{2}+x-2}\)
Step-by-Step Solution
Verified Answer
The function \(f(x) = \frac{2x+1}{(x+2)(x-1)}\) is continuous for all x such that \(x \neq -2\) and \(x \neq 1\).
1Step 1: Factor the Denominator
First, we should factor the denominator to make it easier to solve the equation \(x^2 + x - 2 = 0\). To factor the quadratic, we look for two numbers that multiply to the constant term (-2) and add to the linear term (1).
In this case, the two numbers are 2 and -1. So, the quadratic factors as follows:
\(x^2 + x - 2 = (x + 2)(x - 1)\)
Now the function can be written as:
\(f(x) = \frac{2x+1}{(x+2)(x-1)}\)
2Step 2: Determine the values where the denominator is zero
We need to find the values of x where the denominator of the function, \((x+2)(x-1)\), is equal to 0. We can do this by setting each of the two factors equal to 0 and solving for x:
1) \(x+2 = 0\)
Subtract 2 from both sides:
\(x = -2\)
2) \(x-1 = 0\)
Add 1 to both sides:
\(x = 1\)
3Step 3: Determine the Continuous Domain of the Function
Now we know that the function will not be continuous at the values x = -2 and x = 1, since the denominator is equal to 0 at these points. Therefore, the function is continuous everywhere else.
So the function is continuous for all x such that \(x \neq -2\) and \(x \neq 1\).
Key Concepts
Factoring PolynomialsRational FunctionsDomain of a Function
Factoring Polynomials
The process of factoring polynomials is crucial in identifying the zeros of a polynomial, which are the x-values where the polynomial equals zero. When we factor a quadratic polynomial, like in our exercise with the expression \(x^2 + x - 2\), we're essentially breaking it down into a product of two simpler linear factors.
Factoring can sometimes seem like a puzzle. To find the factors, one approach is to seek two numbers that multiply together to give the constant term (in this case, -2) and add up to the coefficient of the linear term (in this case, 1). Once the polynomial is factored into \(x + 2)(x - 1)\), those factors reveal the points at which the polynomial evaluates to zero; specifically, at the x-values -2 and 1 for this problem.
Understanding how to factor polynomials not only simplifies expressions but also lays the groundwork for higher-level mathematics, such as analyzing the continuity of rational functions—a topic we encounter in the subsequent discussion.
Factoring can sometimes seem like a puzzle. To find the factors, one approach is to seek two numbers that multiply together to give the constant term (in this case, -2) and add up to the coefficient of the linear term (in this case, 1). Once the polynomial is factored into \(x + 2)(x - 1)\), those factors reveal the points at which the polynomial evaluates to zero; specifically, at the x-values -2 and 1 for this problem.
Understanding how to factor polynomials not only simplifies expressions but also lays the groundwork for higher-level mathematics, such as analyzing the continuity of rational functions—a topic we encounter in the subsequent discussion.
Rational Functions
A rational function is a fraction in which both the numerator and the denominator are polynomials. In the example \(f(x)=\frac{2x+1}{x^2+x-2}\), the numerator is \(2x+1\) and the denominator is a quadratic polynomial \(x^2+x-2\) which we have factored. An important chapter in the study of rational functions is the identification of discontinuities.
Rational functions are undefined at x-values that make the denominator zero because division by zero is not allowed in mathematics. These particular x-values are the points at which a rational function may be discontinuous or have asymptotes. Remember that understanding where these functions are continuous or discontinuous directly aids in sketching their graphs, analyzing limits, and solving equations.
Rational functions are undefined at x-values that make the denominator zero because division by zero is not allowed in mathematics. These particular x-values are the points at which a rational function may be discontinuous or have asymptotes. Remember that understanding where these functions are continuous or discontinuous directly aids in sketching their graphs, analyzing limits, and solving equations.
Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions, the domain is usually all real numbers because you can input any x-value without creating mathematical problems. However, for rational functions like \(f(x)=\frac{2x+1}{x^2+x-2}\), we must exclude the x-values that cause the denominator to be zero—these are the x-values where the function cannot exist.
In our exercise, the denominator factors to \(x+2)(x-1)\), so the function is undefined at \(x = -2\) and \(x = 1\). Therefore, the domain of the function is all real numbers except \(x = -2\) and \(x = 1\). Being able to determine the domain of a function is fundamental in both understanding the function's behavior and solving for unknown variables within its context.
In our exercise, the denominator factors to \(x+2)(x-1)\), so the function is undefined at \(x = -2\) and \(x = 1\). Therefore, the domain of the function is all real numbers except \(x = -2\) and \(x = 1\). Being able to determine the domain of a function is fundamental in both understanding the function's behavior and solving for unknown variables within its context.
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