Problem 27 - Algebra 1

Question

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\sqrt{8 x^{2}+\frac{3}{2}} ; \frac{1}{4}$$

Step-by-Step Solution

Answer

The function evaluated at \(x=\frac{1}{4}\) gives \(y=1.4\).

Step 1: Substitute the value \(x=\frac{1}{4}\) into the equation

We can start by substituting the given value of \(x\) which is \(\frac{1}{4}\) into our function: \[y=\sqrt{8(\frac{1}{4})^{2} + \frac{3}{2}}\]

Step 2: Simplify inside the square root

First, calculate \((\frac{1}{4})^{2} = \frac{1}{16}\). Then, compute \(8 (\frac{1}{16}) = 0.5\). Lastly, add the results: \[y=\sqrt{0.5+\frac{3}{2}}\]

Step 3: Further simplify and calculate the square root

Before taking the square root, add \(0.5+(3/2)=2\). Then, compute the square root: \[y=\sqrt{2} ≈ 1.4\]

Key Concepts

Substituting Values

When you encounter a function in mathematics, being able to evaluate it for a specific value is a fundamental skill. This involves substituting a given number for every instance of the variable within the function. To correctly substitute values, it's crucial to follow the order of operations. The function provided in the exercise is a clear demonstration of this concept.

Here's how you do it: Identify the variable in the function, which is represented by the letter 'x', and then replace it with the given value, in this case, \(\frac{1}{4}\). You must be careful to substitute the value everywhere the variable appears. After substitution, the function might look more complex, but remember, it's just a matter of simplifying the expression from there.

Simplifying Expressions

Simplifying expressions is like tidying up a mess; it makes everything clearer and easier to understand. To simplify an expression, combine like terms and perform the operations according to mathematical rules, such as the order of operations—parentheses, exponents, multiplication and division, and addition and subtraction (PEMDAS).

In the given exercise, you first tackle the exponent \(\left(\frac{1}{4}\right)^2\) which simplifies to \(\frac{1}{16}\). Next, you multiply this result by 8, giving you 0.5. The last step includes adding the fraction \(\frac{3}{2}\), which simplifies to 1.5 when expressed as a decimal. All these operations need to be carried out before taking the square root, ensuring the expression under the root is as simple as it can be.

Square Root Calculation

Solving for the square root can be approached in different ways depending on whether you're working with perfect squares or more complex numbers. In the context of our exercise, we're looking at the square root of 2, which is not a perfect square. Therefore, we can't simplify it to a neat whole number or fraction.

In this scenario, we approximate the value using a calculator, rounding it to the nearest tenth as instructed, which gives us approximately 1.4. Understanding how to handle square roots is crucial because they're common in a variety of mathematical fields, from geometry to calculus.