Find the range of the functions:
`f(x)=2-3x-5x^(2)`

Explanation

To find the range of the function \( f(x) = 2 - 3x - 5x^2 \), we can follow these steps:

### Step 1: Rewrite the function
We can rewrite the function in standard form:
\[ f(x) = -5x^2 - 3x + 2 \]

### Step 2: Identify coefficients
In the standard form of a quadratic function \( ax^2 + bx + c \), we identify:
- \( a = -5 \)
- \( b = -3 \)
- \( c = 2 \)

### Step 3: Determine the vertex
The vertex of a parabola given by the equation \( ax^2 + bx + c \) can be found using the formulas:
- \( x = -\frac{b}{2a} \)
- \( y = -\frac{D}{4a} \) where \( D = b^2 - 4ac \)

#### Step 3.1: Calculate \( x \) coordinate of the vertex
\[ x = -\frac{-3}{2 \times -5} = \frac{3}{-10} = -\frac{3}{10} \]

#### Step 3.2: Calculate \( D \)
\[ D = b^2 - 4ac = (-3)^2 - 4(-5)(2) = 9 + 40 = 49 \]

#### Step 3.3: Calculate \( y \) coordinate of the vertex
\[ y = -\frac{D}{4a} = -\frac{49}{4 \times -5} = \frac{49}{20} \]

### Step 4: Identify the nature of the parabola
Since \( a = -5 \) is negative, the parabola opens downwards. This means that the vertex represents the maximum point of the function.

### Step 5: Determine the range
The maximum value of \( f(x) \) is \( \frac{49}{20} \) and as \( x \) approaches positive or negative infinity, \( f(x) \) approaches negative infinity. Therefore, the range of the function is:
\[ (-\infty, \frac{49}{20}] \]

### Final Answer
The range of the function \( f(x) = 2 - 3x - 5x^2 \) is:
\[ (-\infty, \frac{49}{20}] \]

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