Understanding the Function and Its Properties

The function is defined as $f\left(x\right)=2x^3-15x^2+36x+1$, with domain $[0,3]$ and codomain $[1,29]$. We need to determine if it is one-one (injective) and/or onto (surjective).

Step 1: Check if the Function is One-One (Injective)

A function is one-one if every element in the domain maps to a unique element in the codomain. For a continuous function on a closed interval, we can check its monotonicity by finding its derivative.

First, find the derivative: $f^{\prime }\left(x\right)=\frac{d}{dx}(2x^3-15x^2+36x+1)=6x^2-30x+36$.

Set the derivative to zero to find critical points: $6x^2-30x+36=0$.

Divide by 6: $x^2-5x+6=0$.

Factor: $(x-2)(x-3)=0$, so critical points are at $x=2$ and $x=3$.

Now, test the sign of $f^{\prime }\left(x\right)$ in the intervals determined by the critical points within the domain [0, 3]:

• For $x\in [0,2)$, choose $x=1$: $f^{\prime }\left(1\right)=6\left(1\right)-30\left(1\right)+36=6-30+36=12>0$. So, increasing.
• For $x\in (2,3)$, choose $x=2.5$: $f^{\prime }\left(2.5\right)=6\left(6.25\right)-30\left(2.5\right)+36=37.5-75+36=-1.5<0$. So, decreasing.

Since the function increases on [0, 2) and decreases on (2, 3], it is not strictly monotonic over the entire domain. Therefore, it is not one-one. For example, different x-values (like near 0 and near 3) can produce the same f(x) value.

Step 2: Check if the Function is Onto (Surjective)

A function is onto if every element in the codomain [1, 29] is the image of at least one element in the domain [0, 3]. We can check this by finding the range of f(x) on [0, 3].

Evaluate f(x) at the endpoints and critical points:

• At x = 0: $f\left(0\right)=2\left(0\right)-15\left(0\right)+36\left(0\right)+1=1$.
• At x = 2: $f\left(2\right)=2\left(8\right)-15\left(4\right)+36\left(2\right)+1=16-60+72+1=29$.
• At x = 3: $f\left(3\right)=2\left(27\right)-15\left(9\right)+36\left(3\right)+1=54-135+108+1=28$.

The minimum value is 1 (at x=0) and the maximum value is 29 (at x=2). Since f(x) is continuous, by the Intermediate Value Theorem, it takes every value between 1 and 29. The codomain is exactly [1, 29], which matches the range. Therefore, the function is onto.

Final Answer

The function is onto but not one-one.

Related Topics

• One-One Function (Injection): A function where each element of the domain maps to a unique element of the codomain. No two different inputs produce the same output.
• Onto Function (Surjection): A function where every element of the codomain is the image of at least one element from the domain. The range equals the codomain.
• Critical Points: Points where the derivative is zero or undefined. They help in determining local maxima, minima, and monotonicity.
• Intermediate Value Theorem: If a function is continuous on a closed interval [a, b], then it takes every value between f(a) and f(b).

Key Formulae and Theory

• Derivative of a polynomial: $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$.
• Quadratic formula: For $a{x}^2+bx+c=0$, solutions are $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.
• To check monotonicity: If