Problem 52 - College Algebra

Question

Find the domain of each function. $$f(x)=8 x^{2}-5 x+2$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=8x^{2}-5x+2\) is all real numbers.

Step 1: Identify the type of function

The function \(f(x)=8x^{2}-5x+2\) is a quadratic function. This is identified by the highest power of variable \(x\) being 2.

Step 2: Check any restrictions on x

In the function \(f(x)=8x^{2}-5x+2\), there is no division by variable or square root of variable. So, no restrictions exist on the values of \(x\) which means \(x\) can be any real number.

Step 3: State the Domain

Therefore, the Domain of the function \(f(x) = 8x^{2}-5x+2\) is all real numbers.

Key Concepts

Quadratic Functions

Quadratic functions are a special type of polynomial function. They follow the standard form of \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. The key characteristic of quadratic functions is the presence of the \( x^2 \) term, making the highest exponent of \( x \) equal to 2. This gives the function a U-shaped graph known as a parabola.
Parabolas can open upwards or downwards depending on the sign of \( a \) (upwards if \( a > 0 \), downwards if \( a < 0 \)).
If you understand how to interpret these coefficients (\( a \), \( b \), \( c \)), you can sketch the graph and understand its properties, such as its vertex, axis of symmetry, and intercepts.

• **Vertex**: The highest or lowest point of the parabola.
• **Axis of Symmetry**: A vertical line that divides the parabola into two mirror-image halves.

In the given function \( f(x)=8x^{2}-5x+2 \), the highest power of \( x \) is 2, clearly identifying it as a quadratic function.

Real Numbers

Real numbers are all the numbers you can think of that lie along the number line. This includes integers, fractions, rational numbers, irrational numbers, and even zero.
In simpler terms, if you can place a number between any two values on a number line, it is a real number. These numbers are extremely useful in mathematics, especially when dealing with continuous data.

• **Positive Integers**: Numbers like 1, 2, 3.
• **Negative Integers**: Numbers like -1, -2, -3.
• **Fractions and Decimals**: Like \(\frac{1}{2}\), 0.75.
• **Irrational Numbers**: Numbers like \(\pi\), \(\sqrt{2}\) that cannot be expressed as exact fractions.

When finding the domain of a function, we often check if any real numbers can make the function undefined. Functions involving square roots or division by variables might have restrictions, but luckily, quadratic functions like \( f(x)=8x^{2}-5x+2 \) do not. Here, any real number is a valid input.

Function Restrictions

Function restrictions determine what values \( x \) can take in a given function. In algebra, certain operations cause restrictions. Common restrictions occur when you involve division by a variable or square roots.

• **Division by \( x \)**: Causes restrictions since division by zero is undefined.
• **Square Roots**: Demands that whatever is inside the root is non-negative for real-number results.

However, a quadratic function like \( f(x)=8x^{2}-5x+2 \) typically does not involve division by variables or square roots. Therefore, there are no restrictions for \( x \) when determining the domain of this particular function.
Since there aren’t any restrictions, quadratic functions generally have a domain of all real numbers. This means you can substitute any real number for \( x \) without fearing undefined results.