Find whether the following is onto function (surjection) or not.
`f:RtoR` defined by `f(x)=x^(3)+5` for all `x inR`

Answer

To determine whether the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^3 + 5 \) is onto (surjective), we will follow these steps:

### Step 1: Understand the Definition of Onto Function
An onto function (or surjection) is a function where for every element \( y \) in the codomain, there exists at least one element \( x \) in the domain such that \( f(x) = y \).

**Hint:** Remember that for a function to be onto, every possible output (in the codomain) must be achieved by some input (from the domain).

### Step 2: Analyze the Function
The function given is \( f(x) = x^3 + 5 \). This is a cubic function, and we need to determine the range of this function.

**Hint:** Consider the behavior of cubic functions and how they can take on values as \( x \) approaches positive and negative infinity.

### Step 3: Determine the Range of the Function
As \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \) because \( x^3 \) dominates. As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \). Therefore, the function can take any real number value.

**Hint:** You can check specific values of \( x \) to see how \( f(x) \) behaves.

### Step 4: Solve for \( y \)
To check if every real number \( y \) can be expressed as \( f(x) \), we set \( y = x^3 + 5 \) and solve for \( x \):
\[ x^3 = y - 5 \]
\[ x = \sqrt[3]{y - 5} \]
This equation shows that for any real number \( y \), there exists a real number \( x \) such that \( f(x) = y \).

**Hint:** Make sure to note that the cube root function is defined for all real numbers.

### Step 5: Conclusion
Since for every \( y \in \mathbb{R} \), we can find an \( x \in \mathbb{R} \) such that \( f(x) = y \), the function \( f(x) = x^3 + 5 \) is onto.

**Final Answer:** The function \( f(x) = x^3 + 5 \) is onto (surjective).