Problem 13 - Contemporary Precalculus

Question

Refer to these three functions: $$ \begin{aligned} f(x) &=\sqrt{x+3}-x+1 \\ g(t) &=t^{2}-1 \\ h(x) &=x^{2}+\frac{1}{x}+2 \end{aligned} $$ In each case, find the indicated value of the function. $$h(a+k)$$

Step-by-Step Solution

Answer

#Question# Find the value of the function $$h(x) = x^2 + \frac{1}{x} + 2$$ at $$h(a+k)$$, where $$a$$ and $$k$$ are constants.

Step 1: Write down the formula for function h(x)

The given function $$h(x)$$ is: $$h(x) = x^2 + \frac{1}{x} + 2$$

Step 2: Substitute x with (a+k) in the formula

To find the value of $$h(a+k)$$, replace $$x$$ with $$(a+k)$$: $$h(a+k) = (a+k)^2 + \frac{1}{a+k} + 2$$

Step 3: Expand and simplify (a+k)^2

Expand the term $$(a+k)^2$$ to get: $$(a+k)^2 = a^2 + 2ak + k^2$$

Step 4: Combine the terms to obtain the final expression for h(a+k)

Now, plug this expanded form of $$(a+k)^2$$ back into the expression for $$h(a+k)$$: $$h(a+k) = (a^2 + 2ak + k^2) + \frac{1}{a+k} + 2$$ This is the final expression for $$h(a+k)$$.

Key Concepts

Function Evaluation

Understanding function evaluation is crucial for working with mathematical expressions. Function evaluation is the process by which we find the output of a function, given an input.

For example, if we have a function defined as \( f(x) = \frac{1}{x} \) then evaluating \( f(2) \) would involve substituting \( x \) with 2, giving us \( f(2) = \frac{1}{2} \). This process uses the substitution method which is pivotal in function evaluation. In the exercise, when given \( h(a+k) \) it's asking for the output of function \( h \) when \( x \) is replaced with \( a+k \). Evaluating functions often involves further steps to simplify the expression for clarity and practical use.

Substitution Method

The substitution method is a technique in algebra where you replace variables with their corresponding values or other expressions. It's a fundamental skill for evaluating functions, solving equations, and simplifying expressions.

In our exercise, this method is employed when we are asked to find \( h(a+k) \) from the function \( h(x) = x^2 + \frac{1}{x} + 2 \). We substitute \( x \) with \( a+k \), turning the abstract equation into something concrete. Taking care when substituting is essential, particularly when dealing with square terms or fractions, as incorrect substitution can lead to errors in the subsequent evaluation.

Simplifying Expressions

Simplifying expressions is the act of reducing them to their simplest form, making them easier to interpret or further manipulate. This involves expanding brackets, combining like terms, and performing arithmetic operations.

The goal is to make the expression as straightforward as possible without changing its value. In our example, the step of expanding \( (a+k)^2 \) to \( a^2 + 2ak + k^2 \) is part of simplifying the expression. By spreading out the terms, we can easily combine them with other parts of the function or simplify further if possible. Simplification is a vital step in ensuring that function evaluations lead to clear and understandable results.

Algebraic Functions

Algebraic functions are mathematical expressions involving operations like addition, subtraction, multiplication, division, and roots among others, with one or more variables. They play a significant role in various aspects of mathematics and its applications.

The functions \( f(x) \), \( g(t) \), and \( h(x) \) from the provided exercise are examples of algebraic functions. These functions can represent real-world situations or purely abstract concepts in mathematics. When we work with these functions, especially with compositions like evaluating \( h(a+k) \), we apply algebraic manipulation techniques to explore their properties or to find specific outputs.