Problem 80 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(f(-1) = 1\).

Step 1: Identify the given function

The given function is \(f(x) = \sqrt{2x + 3}\).

Step 2: Substitute the input value

Substitute \(x = -1\) into the function \(f(x)\). So you have \(f(-1) = \sqrt{2(-1) + 3}\).

Step 3: Simplify the expression inside the square root

Calculate \(2(-1) + 3\) which equals \( -2 + 3 = 1 \).

Step 4: Calculate the square root

The expression inside the square root becomes \(1\), so \(f(-1) = \sqrt{1}\). Since the square root of 1 is 1, \(f(-1) = 1\).

Key Concepts

Functions

Functions are like mathematical machines. They take an input, process it with a given formula, and produce an output. A function can be of various types, and each type has its unique rule or operation. For example, in the current exercise, the function is defined as \( f(x) = \sqrt{2x + 3} \). This means for any input \( x \), you will follow this expression to get the result or the function's value. Functions are very useful in mathematics as they allow us to encapsulate a set of operations that we can repeat with different inputs.

• Each function has a name, usually denoted as \( f \), \( g \), or another letter.
• Inside the brackets indicates the input variable, usually \( x \).
• The expression following the equals sign shows what happens to \( x \) inside the function.

A key part of understanding functions is knowing what happens at each step from input to output, like how to use the correct expressions and whether certain operations apply, such as square roots or other transformations.

Square Root

The square root is a special mathematical operation that finds the number that, when multiplied by itself, gives the original number. In the expression \( \sqrt{2x + 3} \), the square root operates on \(2x + 3\). Here's what to understand about square roots:

• It's represented by a radical symbol \(\sqrt{}\).
• It only works on non-negative numbers, as square roots of negative numbers aren't real numbers.
• Its purpose is to "undo" the operation of squaring, which means finding the original base number.

In the process of evaluating functions, calculating a square root can change depending on what your inside number becomes after substituting values and simplifying an expression. For example, calculating \( \sqrt{1} \) gives us 1, since 1 squared is 1.

Substitution

Substitution is a simple yet crucial process in mathematics. It's about replacing a variable with a given number. In functions, substitution helps find specific values for the function. Here's how it relates:

• Identify the variable in the function, often \( x \).
• Replace this variable with the value provided - issue solved step by step!
• Calculate the resulting expression, following mathematical rules: order of operations, simplification, etc.

Substitution was key in the exercise where \( x = -1 \) was plugged into \( f(x) \). The resulting \( 2(-1) + 3 \) was simplified to \( 1 \), finally allowing the calculation of \( \sqrt{1} \). Substituting turns an abstract problem into a more concrete one that you can easily solve.

Algebraic Expressions

Algebraic expressions are a central part of math. They combine numbers, variables, and operational symbols to establish a mathematical relationship. In exercises like these, understanding how algebraic expressions work is crucial to solving the problem. Here's what you need to know:

• Variables in expressions can stand for unknowns or applicable numbers.
• Operations like addition, subtraction, multiplication, and division apply to variables and constants.
• Simplification is key. Combine like terms and perform operations within given formulas.

The algebraic expression \( 2x + 3 \) within the function \( f(x) \) highlights how expressions determine function behavior. Knowing how to manipulate and simplify these expressions helps you accurately find the function output for any input. The process involves evaluating each part of the expression according to the defined operations.