Find the image of the following sets under the mapping `f(x)= x^4 -8x^3 +22x^2 -24x + 10` (i) `(-oo,1)`

Answer

To find the image of the set \((-∞, 1)\) under the mapping \(f(x) = x^4 - 8x^3 + 22x^2 - 24x + 10\), we will follow these steps:

### Step 1: Differentiate the function
First, we need to find the derivative of the function \(f(x)\) to determine its critical points.

\[ f'(x) = 4x^3 - 24x^2 + 44x - 24 \]

### Step 2: Set the derivative to zero
Next, we set the derivative equal to zero to find the critical points.

\[ 4x^3 - 24x^2 + 44x - 24 = 0 \]

### Step 3: Factor the derivative
We can factor out a common term from the derivative:

\[ 4(x^3 - 6x^2 + 11x - 6) = 0 \]

Now we can find the roots of the cubic polynomial \(x^3 - 6x^2 + 11x - 6\).

### Step 4: Find the roots of the cubic polynomial
Using the Rational Root Theorem or synthetic division, we can find that \(x = 1\) is a root. We can factor the polynomial as follows:

\[ (x - 1)(x^2 - 5x + 6) = 0 \]

Factoring the quadratic gives us:

\[ (x - 1)(x - 2)(x - 3) = 0 \]

Thus, the critical points are \(x = 1\), \(x = 2\), and \(x = 3\).

### Step 5: Evaluate the function at the critical points
Now we will evaluate \(f(x)\) at these critical points:

1. **For \(x = 1\)**:
\[ f(1) = 1^4 - 8(1^3) + 22(1^2) - 24(1) + 10 = 1 - 8 + 22 - 24 + 10 = 1 \]

2. **For \(x = 2\)**:
\[ f(2) = 2^4 - 8(2^3) + 22(2^2) - 24(2) + 10 = 16 - 64 + 88 - 48 + 10 = 2 \]

3. **For \(x = 3\)**:
\[ f(3) = 3^4 - 8(3^3) + 22(3^2) - 24(3) + 10 = 81 - 216 + 198 - 72 + 10 = 1 \]

### Step 6: Determine the behavior of \(f(x)\) in the interval
Now we need to analyze the behavior of \(f(x)\) as \(x\) approaches \(-\infty\) and at the critical points:

- As \(x \to -\infty\), \(f(x) \to +\infty\).
- At \(x = 1\), \(f(1) = 1\).
- At \(x = 2\), \(f(2) = 2\).
- At \(x = 3\), \(f(3) = 1\).

### Step 7: Find the image of the interval \((-∞, 1)\)
Since \(f(x)\) approaches \(+\infty\) as \(x\) approaches \(-\infty\) and decreases to \(1\) at \(x = 1\), the image of the interval \((-∞, 1)\) under \(f(x)\) is:

\[ (1, +\infty) \]

### Final Answer
The image of the set \((-∞, 1)\) under the mapping \(f(x)\) is \((1, +\infty)\).

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