Problem 82 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value of \( f(3) \) is 3.

Step 1: Understand the Given Function

We are given the function \( f(x) = \sqrt{2x + 3} \). We need to evaluate this function at \( x = 3 \) to find \( f(3) \).

Step 2: Substitute into the Function

Substitute \( x = 3 \) into the function \( f(x) = \sqrt{2x + 3} \). This yields: \( f(3) = \sqrt{2(3) + 3} \).

Step 3: Calculate Inside the Square Root

Calculate the expression inside the square root: \( 2(3) + 3 = 6 + 3 = 9 \).

Step 4: Solve the Square Root

Find the square root of the result from Step 3: \( \sqrt{9} = 3 \).

Step 5: Conclusion: Final Function Value

The value of the function \( f(3) \) is \( 3 \).

Key Concepts

Function Evaluation

Evaluating a function means finding the output value of that function for a specific input value. If you have a function, say \( f(x) \), evaluating \( f(3) \) involves replacing \( x \) in the function with 3 and then simplifying to get an answer. Function evaluation is a critical skill in algebra because it shows how functions transform or map inputs to outputs.

• Think of a function as a mathematical machine that takes an input, does something to it, and then gives you an output.
• In our example, we need to evaluate \( f(3) \) which requires substituting 3 into the function \( f(x)=\sqrt{2x+3} \).
• This involves straightforward substitution followed by basic mathematical operations like addition and finding square roots.

Understanding function evaluation helps in graphing functions, solving equations, and modeling real-life situations where functions describe relationships between variables.

Square Roots

The square root is a special mathematical function that, when applied to a number, gives the value which, when multiplied by itself, results in the original number. For instance, the square root of 9 is 3 because 3 multiplied by itself is 9.

• In mathematical terms, the square root of a number \( y \) is a number \( x \) such that \( x^2=y \).
• Using the radical symbol \( \sqrt{} \), we express it as \( \sqrt{y} = x \).
• For positive integer outcomes, square roots are relatively simple, but finding square roots of non-perfect squares requires a more complex approach.

In the exercise, computing \( \sqrt{9} \) gives us the value 3, a critical last step in evaluating our function, \( f(3) \). Understanding square roots is essential for working effortlessly with quadratic equations, geometry, and various algebraic contexts where inverse operations are required.

Substitution in Functions

Substitution is a vital technique used to evaluate functions and solve algebraic expressions by replacing a variable with a specific value or another expression.

• When substituting into a function, you literally "switch out" the variable – in our case \( x \) – with a given number or term.
• This allows us to find specific outputs of the function, also known as function values, for particular inputs.
• In the given exercise, substituting \( x = 3 \) involved replacing \( x \) in \( f(x)=\sqrt{2x+3} \) to compute \( f(3) \).

The power of substitution lies in its simplicity, offering a straightforward way to manipulate and explore functions. It is a foundational concept in algebra that not only helps in evaluating functions but also in simplifying expressions and solving equations.