Problem 64 - College Algebra Essentials

Question

Let $$\begin{aligned}&f(x)=2 x-5\\\&g(x)=4 x-1\\\&h(x)=x^{2}+x+2\end{aligned}$$ Evaluate the indicated function without finding an equation for the function. $$f(g[h(1)])$$

Step-by-Step Solution

Answer

The value of \(f(g(h(1)))\) is 25.

Step 1: Substitute x into h(x)

First, let's substitute \(x=1\) into \(h(x) = x^2 + x + 2\). This yields \(h(1) = 1^2 + 1 + 2 = 4\).

Step 2: Substitute the outcome into g(x)

Next, substitute \(h(1) = 4\) into \(g(x) = 4x - 1\). This gives \(g(h(1)) = g(4) = 4*4 - 1 = 15\).

Step 3: Substitute the result into f(x)

Finally, substitute \(g(h(1)) = 15\) into \(f(x) = 2x - 5\). This results in \(f(g(h(1))) = f(15) = 2*15 - 5 = 25\).

Key Concepts

Composite Functions

Understanding composite functions is integral to higher-level algebra. A composite function is essentially one function nested within another. This concept is frequently represented as \( f(g(x)) \), which can be read as 'f of g of x'. To evaluate a composite function, you start from the innermost function and progress outward. It's like peeling an onion, layer by layer, to get to the core.

In the given exercise, we compute \( f(g(h(1))) \) without directly finding the equation for the composite function. Step by step, we substitute and simplify using each function's definition until we reach the final value. The concept of composite functions is ubiquitous in mathematics and can represent more complex relationships between variables.

Function Evaluation

Function evaluation is the process of determining the output of a function for a particular input. To evaluate a function, you replace its variable with a given number and perform the arithmetic operations as indicated by the function's formula.

For instance, in our exercise, evaluating \( h(1) \) means we substitute \( x \) with \( 1 \) in the function \( h(x) = x^2 + x + 2 \). After the substitution, we calculate the arithmetic to find that \( h(1) \) equals \( 4 \). This process is crucial for understanding how functions behave and how changes in input values affect the output.

Algebraic Functions

Algebraic functions are functions that can be expressed using algebraic expressions, involving operations like addition, subtraction, multiplication, division, and taking roots among constants and variables. The functions \( f(x) \) and \( g(x) \) from our exercise are algebraic functions, as they involve linear combinations of the variable \( x \).

Algebraic functions are the bread and butter of algebra and are used to model real-world phenomena. By manipulating these functions, we can derive meaningful insights into the behaviour of different systems described by the functions.

Function Operations

In algebra, function operations include addition, subtraction, multiplication, division, and composition. These operations allow mathematicians to combine and transform functions, leading to more complex or simplified functions as needed.

In the given exercise, while we don't perform standard operations on functions like addition or subtraction, we do apply function composition, which is an advanced operation. This involves substituting the output of one function directly into another. It highlights the importance of understanding both the algebraic manipulation and the conceptual impact of function operations in mathematics.