The Values of 'a' for Which the Function f(x) = x^3 + x^2 + a cos(x) is Injective

What does it mean for a function to be injective? This is where we begin: when considering a function, injectivity implies that the function gives unique outputs for distinct inputs. This uniqueness is key in determining how the function behaves under different conditions.

Now, let’s focus on the specific function given:
$f\left(x\right)=x^3+x^2+a\cos \left(x\right)$f(x)=x3+x2+acos(x)
Where 'a' is a constant parameter. The task is to find for which values of 'a', this function becomes injective. Injectivity of a function means that if $f\left(x_1\right)=f\left(x_2\right)$f(x1​)=f(x2​), then $x_1=x_2$x1​=x2​. This condition can be rewritten in terms of our function:
$x_1^3+x_1^2+a\cos \left(x_1\right)=x_2^3+x_2^2+a\cos \left(x_2\right)$x13​+x12​+acos(x1​)=x23​+x22​+acos(x2​)
For injectivity to hold, this must imply $x_1=x_2$x1​=x2​.

The Role of the Cosine Function

One of the most challenging components of this function is the cosine term. The cosine function is periodic and bounded, meaning that no matter what value of $x$x is chosen, $\cos \left(x\right)$cos(x) will always lie between -1 and 1, and it repeats its values every $2\pi$2π units. This periodicity can lead to multiple values of $x$x producing the same $\cos \left(x\right)$cos(x), which could disrupt injectivity unless the cubic and quadratic terms offset this behavior.

Graphical Analysis:

To gain a deeper understanding, let’s analyze the behavior of each term separately:

1. $x^3$x3: This cubic term is strictly increasing for all values of $x$x, which means it contributes positively towards making the function injective.
2. $x^2$x2: The quadratic term, however, is not injective by itself because it is symmetric around $x=0$x=0. For example, $f\left(1\right)=f(-1)$f(1)=f(−1), and thus this term could potentially introduce non-injectivity.
3. $a\cos \left(x\right)$acos(x): This term, as mentioned earlier, oscillates between -1 and 1. If the amplitude 'a' is large enough, it could dominate the behavior of the cubic and quadratic terms, making the function non-injective.

Critical Insight: Balancing the Terms

The injectivity of the overall function depends on the balance between these terms. For very large values of 'a', the cosine term could dominate, causing the function to fail injectivity due to the periodic nature of cosine. On the other hand, for very small values of 'a', the cubic term would dominate, and the function could potentially remain injective because the cubic term is strictly increasing.

First Case: Large Values of 'a'

Let’s first consider what happens when $a$a is large. If $a$a becomes sufficiently large, the cosine term introduces multiple intervals of non-injectivity due to its periodicity. For example, $\cos \left(x\right)$cos(x) will repeat its values every $2\pi$2π interval, meaning that at some points $f\left(x_1\right)=f\left(x_2\right)$f(x1​)=f(x2​) even though $x_1\ne x_2$x1​=x2​.

Second Case: Small or Zero Values of 'a'

When $a=0$a=0, the function simplifies to $f\left(x\right)=x^3+x^2$f(x)=x3+x2. This is a combination of cubic and quadratic terms, both of which are polynomials. While the quadratic part might seem concerning due to its symmetry, the cubic term dominates as $x\to \pm \infty$x→±∞, and thus the function remains injective.

For small positive or negative values of $a$a, the cosine term provides only a small perturbation to the cubic and quadratic terms. In this case, the function is likely to remain injective because the dominant cubic term grows rapidly enough to offset any oscillations introduced by the cosine term.

Conclusion:

Based on this analysis, we conclude that the function $f\left(x\right)=x^3+x^2+a\cos \left(x\right)$f(x)=x3+x2+acos(x) remains injective for sufficiently small values of $a$a. More precisely, if $\mid a\mid$∣a∣ is small enough, the cubic term ensures that the function remains strictly increasing and, thus, injective. However, for larger values of $a$a, the periodic nature of the cosine term will likely introduce points where injectivity is lost.

This conclusion can be visually verified by plotting the function for different values of $a$a. A table summarizing the behavior of the function for different values of $a$a is provided below:

Value of $a$a	Behavior of $f\left(x\right)$f(x)	Injective?
$a=0$a=0	$f\left(x\right)=x^3+x^2$f(x)=x3+x2	Yes
Small $a$a	Dominated by cubic term	Yes
Large $a$a	Dominated by cosine term	No

In summary, the function $f\left(x\right)=x^3+x^2+a\cos \left(x\right)$f(x)=x3+x2+acos(x) is injective when the constant $a$a is small. Larger values of $a$a introduce periodicity that disrupts the injectivity. Understanding this behavior is crucial when dealing with such mixed polynomial and trigonometric functions.