Problem 63 - 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. $$g(f[h(1)])$$

Step-by-Step Solution

Answer

The evaluation of \( g(f(h(1))) \) results in 11.

Step 1: Evaluate h(1)

To evaluate \( h(1) \), we replace \( x \) with 1 in the equation of \( h(x) \). Thus, \( h(1) = 1^2 + 1 + 2 \). This simplifies to 4.

Step 2: Evaluate f(h(1))

Next, we evaluate \( f(h(1)) \), which means substituting \( h(1) \) (which we previously found to be 4) into \( f(x) = 2x - 5 \). Thus, \( f(h(1)) = 2*4 - 5 \), which simplifies to 3.

Step 3: Evaluate g(f(h(1)))

Lastly, we evaluate \( g(f(h(1))) \), which means substituting \( f(h(1)) \) (which we previously found to be 3) into \( g(x) = 4x - 1 \). Thus, \( g(f(h(1))) = 4*3 - 1 \), which simplifies to 11.

Key Concepts

Function Composition

Function composition involves combining two or more functions in such a way that the output of one function becomes the input of the next. It's like a relay race where the handoff of the baton is the output of one racer (function) becoming the input for the next. For example, if we have two functions, say, f and g, the composition of f with g, denoted as g(f(x)), involves evaluating f first and then using its result as the input for g.

Visualizing this with our textbook example, we first find the value of h(x) for a specific value of x, then we use that result as the input for f(x), and finally, the result from f(x) becomes the input for g(x). This process shows how function composition chains together in a specific order that cannot be interchanged without changing the overall result.

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. It's like using a vending machine where you select an option (input) and receive an item (output). To evaluate a function, we substitute the input value into the function's formula. For instance, in the function h(x) = x^2 + x + 2, evaluating h(1) means replacing every x with 1 which results in h(1) = 1^2 + 1 + 2, simplifying further to h(1) = 4.

The textbook example demonstrates this through a sequence of evaluations, starting with h(1), then f(h(1)), and finally g(f(h(1))). Each step requires careful substitution and simplification to ensure accuracy. Proper evaluation is crucial when dealing with composite functions since each step builds upon the previous one.

Algebraic Functions

Algebraic functions are mathematical expressions that involve operations like addition, subtraction, multiplication, division, and raising to powers with algebraic terms. These functions are the building blocks of algebra and are crucial for understanding relationships between variables. For example, the functions f(x) = 2x - 5, g(x) = 4x - 1, and h(x) = x^2 + x + 2 from our exercise are all algebraic functions. They are defined using polynomials or differences of polynomials,

and every function has a specific rule that applies to any input within its domain. The power of algebraic functions lies in their versatility; they can model a vast array of real-world phenomena and are foundational in most fields of science and engineering. Understanding how to manipulate these functions is essential for solving more complex problems in mathematics.