Problem 31
Question
For what values of \(a\), the function \(f(x)=\left(a^{2}+a-2\right) x+a^{2}+2 a-3\) is (a) even, (b) odd?
Step-by-Step Solution
Verified Answer
The values for which the function is (a) even are \(a = -2\) and \(a = 1\), and for (b) odd, there is no solution.
1Step 1: Substitute \(-x\) for \(x\) in the function
Replace every instance of \(x\) with \(-x\) in the given function:
\(f(-x) = (a^2+a-2)(-x) + a^2 + 2a - 3\)
2Step 2: Simplify the expression for \(f(-x)\)
Take the negative sign out of the parentheses and simplify:
\(f(-x) = -(a^2+a-2)x + a^2 + 2a - 3\)
3Step 3: Determine when \(f(-x) = f(x)\) (even function)
Compare this with the original function \(f(x) = (a^2+a-2) x + a^2 + 2a - 3\). We can see that the expressions are equal if:
\(-(a^2+a-2)x + a^2 + 2a - 3 = (a^2+a-2) x + a^2 + 2a - 3\)
From this, we have
\((-1)(a^2+a-2) = a^2+a-2\)
Solve for \(a\):
\(-a^2 - a + 2 = a^2 + a - 2\)
\(2a^2 + 2a - 4 = 0\)
\(a^2 + a - 2 = 0\) (divide both sides by 2)
4Step 4: Solve the quadratic equation for the even function
Factor the quadratic equation in step 3:
\((a+2)(a-1) = 0\)
Finally, solve for values of \(a\):
\(a = -2\) or \(a = 1\)
These are the values of \(a\) which make the function even.
5Step 5: Determine when \(f(-x) = -f(x)\) (odd function)
Now, we want to find the values of \(a\) for which the function is odd. So, we set:
\(-(a^2+a-2)x + a^2 + 2a - 3 = -[(a^2+a-2) x + a^2 + 2a - 3]\)
The left side is already equal to \(f(-x)\), so we need to evaluate the right side:
\(-a^2 - a + 2)x - a^2 - 2a + 3\)
Now, we set \(f(-x)\) equal to \(-f(x)\):
\(-(a^2+a-2)x + a^2 + 2a - 3 = (-a^2 - a + 2)x - a^2 - 2a + 3\)
6Step 6: Analyze the coefficients of \(x\) in the odd function
Looking at the coefficients of \(x\) on both sides of the equation, we see that they are always equal:
\(-(a^2+a-2) = -a^2 - a + 2\)
Since the coefficients are always equal, there is no solution for \(a\) that makes the function odd.
#Conclusion#:
- even function: \(a = -2\) or \(a = 1\)
- odd function: no solution
Key Concepts
Quadratic EquationsFunction AnalysisSymmetry in Functions
Quadratic Equations
Quadratic equations are a fundamental concept in algebra. A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The solutions to quadratic equations are often referred to as roots, and they can be found using various methods such as factoring, completing the square, or using the quadratic formula. Understanding quadratic equations includes analyzing their graphs, which are parabolas. These parabolas can open upwards or downwards depending on the coefficient \(a\).
In the context of our exercise, the quadratic equation arises when we compare expressions to determine when a function is even. We simplified and factored the equation \(a^2 + a - 2 = 0\) to find values of \(a\) that satisfy the condition for the function \(f(x)\) to be even, yielding \(a = -2\) or \(a = 1\). This shows how quadratic equations are used in function analysis to identify specific properties, like symmetry.
In the context of our exercise, the quadratic equation arises when we compare expressions to determine when a function is even. We simplified and factored the equation \(a^2 + a - 2 = 0\) to find values of \(a\) that satisfy the condition for the function \(f(x)\) to be even, yielding \(a = -2\) or \(a = 1\). This shows how quadratic equations are used in function analysis to identify specific properties, like symmetry.
Function Analysis
Function analysis involves examining the properties and behaviors of a function. When analyzing functions, it's essential to look at characteristics such as domain, range, intercepts, and behavior as \(x\) approaches infinity. Another critical aspect of function analysis is determining whether a function is even or odd.
For our specific problem, function analysis includes substituting \(-x\) for \(x\) to test the function's symmetry. By defining \(f(-x)\) and comparing it with \(f(x)\), we determine whether the function is even (\(f(-x) = f(x)\)) or odd (\(f(-x) = -f(x)\)).
During this process, algebraic manipulation plays a key role. Simplifying the resulting expressions and analyzing the coefficients helps us find solutions for \(a\) that satisfy requirements for symmetry. This thorough analytical approach is a cornerstone of function analysis, allowing us to ascertain specific characteristics of the function.
For our specific problem, function analysis includes substituting \(-x\) for \(x\) to test the function's symmetry. By defining \(f(-x)\) and comparing it with \(f(x)\), we determine whether the function is even (\(f(-x) = f(x)\)) or odd (\(f(-x) = -f(x)\)).
During this process, algebraic manipulation plays a key role. Simplifying the resulting expressions and analyzing the coefficients helps us find solutions for \(a\) that satisfy requirements for symmetry. This thorough analytical approach is a cornerstone of function analysis, allowing us to ascertain specific characteristics of the function.
Symmetry in Functions
Symmetry in functions is a crucial concept in mathematics, which helps simplify the analysis of different types of functions. Symmetry can be visually observed in a function's graph and understood through algebraic expressions.
For our exercise, we were tasked with determining whether the function \(f(x) = (a^2 + a - 2)x + a^2 + 2a - 3\) is even or odd. A function is **even** if its graph is symmetrical with respect to the y-axis. This means \(f(x) = f(-x)\) for all \(x\). If \(a = -2\) or \(a = 1\), our function satisfies this condition and is thus even.
A function is **odd** if its graph has rotational symmetry around the origin, implying \(f(x) = -f(-x)\). However, as we analyzed earlier, no value of \(a\) makes the function odd in this exercise.
Understanding these symmetrical properties allows mathematicians and students alike to draw significant conclusions about functions quickly, which aids in further mathematical exploration or application.
For our exercise, we were tasked with determining whether the function \(f(x) = (a^2 + a - 2)x + a^2 + 2a - 3\) is even or odd. A function is **even** if its graph is symmetrical with respect to the y-axis. This means \(f(x) = f(-x)\) for all \(x\). If \(a = -2\) or \(a = 1\), our function satisfies this condition and is thus even.
A function is **odd** if its graph has rotational symmetry around the origin, implying \(f(x) = -f(-x)\). However, as we analyzed earlier, no value of \(a\) makes the function odd in this exercise.
Understanding these symmetrical properties allows mathematicians and students alike to draw significant conclusions about functions quickly, which aids in further mathematical exploration or application.
Other exercises in this chapter
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