Problem 31 - Algebra for College Students

Question

If \(f(x)=\sqrt{x}\) and \(g(x)=3 x-1\), find \((f \circ g)(4)\) and \((g \circ f)(4) . \quad \sqrt{11} ; 5\)

Step-by-Step Solution

Answer

\( (f \circ g)(4) = \sqrt{11} \); \( (g \circ f)(4) = 5 \).

Step 1: Understand Function Composition

Function composition involves applying one function to the result of another. In particular, for two functions \(f(x)\) and \(g(x)\), \((f \circ g)(x)\) means you first apply \(g\) to \(x\), and then apply \(f\) to the resulting value.

Step 2: Compute \(g(4)\)

For the function \(g(x) = 3x - 1\), substitute \(x = 4\): \[ g(4) = 3 \times 4 - 1 = 12 - 1 = 11 \]

Step 3: Apply \(f") on \(g(4) = 11\)

Use the result from Step 2 in the function \(f(x) = \sqrt{x}\): \[ f(g(4)) = f(11) = \sqrt{11} \]

Step 4: Compute \(f(4)\)

For the function \(f(x) = \sqrt{x}\), substitute \(x = 4\): \[ f(4) = \sqrt{4} = 2 \]

Step 5: Apply \(g\) on \(f(4) = 2\)

Use the result from Step 4 in the function \(g(x) = 3x - 1\): \[ g(f(4)) = g(2) = 3 \times 2 - 1 = 6 - 1 = 5 \]

Key Concepts

Composite Functions

Function composition is a fundamental concept in mathematics that involves combining two or more functions to create a new function. This process is much like layering processes or tasks to achieve a result. Think of it as one function's output becoming the next function's input. An easy way to visualize this is by imagining a conveyor belt. Each stage of the belt modifies the item slightly until a final product is achieved. In math, instead of an item, it's a number or an expression. For our given example, if we have two functions, \( f(x) = \sqrt{x} \) and \( g(x) = 3x - 1 \), and we need to find \( (f \circ g)(4) \). The notation \( (f \circ g)(x) \) means perform \( g(x) \) first, then apply \( f \) to the result of \( g(x) \). This commonly involves two steps:

• Calculate \( g(4) \) first, which gives us \( 11 \), because \( 3 \times 4 - 1 = 11 \).
• Next, use this result in function \( f \), so \( f(11) = \sqrt{11} \).

Function Evaluation

Function evaluation involves determining the output of a function for a specific input. It is the process of substituting a particular value into the function's formula to find the corresponding result. This is like a recipe where you input ingredients to get a dish; your input leads directly to an output. To evaluate a function:

• Identify your input value \( x \).
• Substitute this \( x \) into your given function equation.
• Simplify the result to get the output.

In our problem, to evaluate \( g(x) = 3x - 1 \) at \( x = 4 \), you plug 4 into the equation: \( 3 \times 4 - 1 = 11 \). Similarly, evaluating \( f(x) = \sqrt{x} \) at \( x = 11 \), gives us \( \sqrt{11} \). The same method applies for carrying out \( (g \circ f)(4) \), where you compute \( f(4) \) firstly, resulting in \( 2 \), and then substitute this into \( g \): \( 3 \times 2 - 1 = 5 \).

Algebra Functions

Understanding algebra functions is key to mastering composite functions and function evaluation. Algebra functions are mathematical expressions involving variables (like \( x \)) and constants combined with arithmetic operations such as addition, subtraction, multiplication, and division. In our example, the functions \( f(x) \) and \( g(x) \) are algebraic functions:

• \( f(x) = \sqrt{x} \) involves a square root, which is a type of exponent function typically seen in algebra.
• \( g(x) = 3x - 1 \) is a linear function involving multiplication and subtraction.

When dealing with algebra functions, it is important to understand the domain and range of the functions, which refer to the set of possible input values (domain) and the possible outputs (range). For instance, \( f(x) = \sqrt{x} \) only works with non-negative numbers because you cannot square root a negative number without involving complex numbers. Such comprehension assists in accurate function composition and evaluation. Mastering these algebraic functions forms a strong foundation for tackling more complex mathematical concepts, enhancing both your problem-solving abilities and mathematical fluency.