Problem 20 - 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. $$g(1-r)$$

Step-by-Step Solution

Answer

Answer: The indicated value of the function \(g(1-r)\) is \(r^2 - 2r\).

Step 1: Write down the given function for g(t)

We have the function \(g(t) = t^2 - 1\).

Step 2: Replace t with (1-r) in the function g(t)

To find \(g(1-r)\), we substitute \(t\) with \((1-r)\) in the formula for \(g(t)\). We have: $$ g(1-r) = (1-r)^2 - 1 $$

Step 3: Expand and simplify the expression

Now, we expand and simplify the expression for \(g(1-r)\): $$ \begin{aligned} g(1-r) &= (1-r)^2 - 1 \\ &= (1 - 2r + r^2) - 1 \\ &= 1 - 2r + r^2 - 1 \\ &= r^2 - 2r \end{aligned} $$

Step 4: Write down the final expression for g(1-r)

We found that: $$ g(1-r) = r^2 - 2r $$ This is the indicated value of the function \(g(1-r)\).

Key Concepts

Function Substitution

When we talk about function substitution, it means replacing the independent variable with another expression. For example, the function \( g(t) = t^2 - 1 \) becomes \( g(1-r) \) when we substitute \( t \) with \( 1-r \). This process helps us understand how changing the input of the function affects the output.
This technique is particularly useful because it allows us to explore the behavior of a function under different conditions. You start by taking the original function's rule, then replace all instances of the variable with the new expression. Make sure to perform this substitution carefully to ensure accuracy.
Function substitution is like changing the perspective of a problem, allowing us to analyze it from different angles.

Polynomial Expansion

Polynomial expansion refers to the process of multiplying out expressions that involve powers, like \((1-r)^2\). The goal is to express the polynomial in a simplified, expanded form.
To expand \((1-r)^2\), apply the distributive property:

• Square the first term: \(1^2 = 1\)
• Square the second term: \((-r)^2 = r^2\)
• Multiply both terms once as a product of two different terms, then double it: \(2 \times 1 \times (-r) = -2r\)

This results in: \[1 - 2r + r^2\] Expanding polynomials is an essential skill because it simplifies the problem, making it much easier to handle in subsequent steps.

Algebraic Simplification

Once you've expanded a polynomial, the next step is algebraic simplification. This involves combining like terms to make the expression easier to read and solve. Let's look at how this works with our example:

• The expanded form of \(g(1-r)\) is \(1 - 2r + r^2 - 1\).
• Combine the constant terms: \(1 - 1 = 0\), so they cancel each other out.
• This leaves us with \(r^2 - 2r\).

Algebraic simplification reduces complex expressions to manageable forms. Simplifying makes it easier to identify relationships or solve equations, focusing only on the necessary components.

Quadratic Expressions

Quadratic expressions include any polynomial where the highest power of the variable is 2. They often appear in the form \(ax^2 + bx + c\). In the function we've been handling, \(g(1-r) = r^2 - 2r\), we indeed have a quadratic expression.

• \(a = 1\) because it's a coefficient of \(r^2\).
• \(b = -2\) as it accompanies \(r\).
• \(c = 0\) since there's no constant term left.

Understanding the nature of quadratic expressions is vital because their structure is used widely, from graphing parabolas to solving quadratic equations using various methods like factoring, completing the square, or the quadratic formula. They form a cornerstone in algebra for their diverse applications in mathematics and science.