Problem 86 - Algebra A Combined Function

Question

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ f(-1) $$

Step-by-Step Solution

Answer

The value of \( f(-1) \) is \( 1 \).

Step 1: Understanding the Function

We begin by identifying the function we need to evaluate. The function given is \( f(x)=\sqrt{2x+3} \). This means we will substitute the given value of \( x \) into this square root function.

Step 2: Substitute the Given Value

Now, substitute \( x = -1 \) into the function: \( f(-1) = \sqrt{2(-1) + 3} \).

Step 3: Calculate the Expression Inside the Square Root

Perform the arithmetic operation inside the square root: \( 2(-1) + 3 = -2 + 3 = 1 \).

Step 4: Evaluate the Square Root

Now, take the square root of the result obtained: \( \sqrt{1} = 1 \).

Key Concepts

Square Root Function

A square root function is a type of mathematical expression that involves the square root of a variable, typically expressed as \( f(x) = \sqrt{x} \). The square root symbol, \(\sqrt{\cdot}\), represents a value that, when multiplied by itself, would equal the number under the square root sign. For instance, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \). In the function \( f(x) = \sqrt{2x + 3} \), the expression inside the square root symbol is known as the radicand (in this case, \( 2x + 3 \)). Evaluating the square root function means calculating the square root of this radicand after substituting the given value of \( x \). Remember that the square root function is only defined for non-negative radicands in the realm of real numbers, ensuring that their values remain real.

Substituting Values

Substituting values involves replacing the variable in a function with a given number to evaluate the expression. This is a crucial step in evaluating any function, as it allows you to assess the function's behavior for specific inputs. To substitute a value into the function \( f(x) = \sqrt{2x + 3} \), you replace \( x \) with a given number. For example, if we need to find \( f(-1) \), we substitute \(-1\) for \( x \) in the function: \( f(-1) = \sqrt{2(-1) + 3} \). After substituting, carry out any arithmetic operations inside the square root. It is essential to carefully follow the order of operations, starting with multiplication or division before addition or subtraction.

Arithmetic Operations

Arithmetic operations involve performing basic math functions such as addition, subtraction, multiplication, and division. In the context of evaluating functions, especially those involving square roots, these operations play a critical role. For the expression \( 2(-1) + 3 \) within the function \( f(-1) = \sqrt{2(-1) + 3} \), you first multiply the numbers: \( 2 \times (-1) = -2 \).
Then, perform the addition: \(-2 + 3 = 1\). After simplifying the expression inside the square root, you evaluate its square root: \( \sqrt{1} = 1 \).

Performing arithmetic operations accurately is crucial for correctly evaluating the square root and any function in general. Follow the order of operations—often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)—to ensure each step is executed correctly.