Problem 20 - Precalculus : Building Concepts and Connections

Question

Use \(f(x)=3 \ln x-4\). Evaluate \(f(1)\).

Step-by-Step Solution

Answer

The value of \(f(1)\) is -4.

Step 1: Understand function notation

The function \(f(x) = 3 \ln x - 4\) is an expression involving the variable x. When we say \(f(1)\), we are replacing 'x' in the function with '1'.

Step 2: Substitute the value of x in the expression

Replace 'x' with '1' in the expression. So, \(f(1) = 3 \ln 1 - 4\).

Step 3: Evaluate the expression

\(\ln 1 = 0\), because any number raised to the power 0 equals 1 . So \(f(1) = 3 * 0 - 4 = -4\).

Key Concepts

Function Evaluation

Understanding function evaluation is crucial when it comes to working with mathematical functions. At its core, function evaluation is the process of finding the output of a function for a specific input value. In simpler terms, whenever you see an expression like \( f(1) \), you need to substitute the given input (in this case, 1) into the function.

• Functions are like machines; you feed in a number (input), and they give you a new number (output).
• The given function in this exercise is \( f(x) = 3 \ln x - 4 \).
• To evaluate \( f(1) \), substitute 1 for \( x \) in the function's formula.

By understanding these concepts, you effectively learn how to navigate through different function evaluations, whether they be linear, quadratic, or even logarithmic ones.

Natural Logarithms

Natural logarithms are a special type of logarithm with the base \( e \), where \( e \approx 2.71828 \). They are often denoted as \( \ln x \) in mathematics. Natural logarithms are a fundamental concept in calculus and appear in many fields like economics, physics, and engineering.

• The natural logarithm of a number is the power to which \( e \) must be raised to obtain that number.
• Importantly, \( \ln 1 = 0 \), because any number raised to the power of 0 equals 1.
• This property is frequently used to simplify expressions, as seen in our exercise: \( \ln 1 \) resulted in 0.

Grasping the basics of natural logarithms allows you to solve complex equations by simplifying exponential growth or decay models found in various scientific contexts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication). They form the basis for equations and functions, letting us solve for unknown values. In our exercise \( f(x) = 3 \ln x - 4 \), this is an algebraic expression concerning the variable \( x \).

• The term \( 3 \ln x \) represents a multiplication of the logarithmic function \( \ln x \) with the coefficient 3.
• The expression ends with the constant term -4, which modifies the result of the logarithmic part by subtracting 4.
• By substituting \( x \) with a specific value, like we did with \( x = 1 \), you can simplify it to a single number.

Mastering algebraic expressions aids in solving more complex algebraic equations and understanding various mathematical models.