Problem 19 - 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. $$g(s+1)$$

Step-by-Step Solution

Answer

Answer: The expression for $$g(s+1)$$ is $$g(s+1) = s^2 + 2s$$.

Step 1: Identify the given function

We are asked to find the value of $$g(s+1)$$, so we will be focusing on the function $$g(t) = t^2 - 1$$.

Step 2: Substitute the variable t with (s+1) in the function g(t)

Now we will substitute the variable t with (s+1) in the function $$g(t) = t^2 - 1$$, which will give us $$g(s+1) = (s+1)^2 - 1$$.

Step 3: Simplify the expression g(s+1)

Next, we will simplify the expression $$g(s+1) = (s+1)^2 - 1$$ by expanding the square and combining like terms: $$ \begin{aligned} g(s+1) &= (s+1)^2 - 1 \\ &= (s^2 + 2s+ 1) - 1 \\ &= s^2 + 2s \end{aligned} $$

Step 4: Write the final expression for g(s+1)

The simplified expression for $$g(s+1)$$ is $$g(s+1) = s^2 + 2s$$. This is the final expression representing the value of the function g(s+1).

Key Concepts

Substitution in Functions

Substitution in functions is a fundamental concept in algebra. It involves replacing a variable in a function with another value or expression. In this exercise, we replaced the variable \( t \) in the function \( g(t) = t^2 - 1 \) with \( (s+1) \). Let's break down the steps:

• Identify the function: First, we focus on the specific function involved. In this case, it's \( g(t) \).
• Perform the substitution: Replace the variable \( t \) with \( s+1 \) to find \( g(s+1) \). This results in the expression \( (s+1)^2 - 1 \).

Understanding substitution helps us manipulate and analyze functions efficiently by altering inputs and observing changes in outputs.

Simplifying Expressions

After substituting a variable in a function, simplifying the expression is essential. It means reducing it to its most basic form so that it is easier to understand or use for further calculations.When we substituted \( s+1 \) into \( g(t) \), we got \( (s+1)^2 - 1 \). Here's how we simplify:

• Expand: Calculate \( (s+1)^2 \) to get \( s^2 + 2s + 1 \).
• Subtract: Remove 1 from the expanded result, giving us \( s^2 + 2s + 1 - 1 \).
• Combine like terms: Simplifies to \( s^2 + 2s \).

Breaking down expressions into smaller parts makes them easier to work with and solves problems efficiently. This skill becomes particularly useful in algebra.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers, usually added or subtracted. They have the general form \( a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1 x + a_0 \).In our exercise, \( g(s+1) = s^2 + 2s \) is a polynomial. Polynomials are categorized based on the highest power of the variable:

• Degree: The degree of a polynomial is the highest power of the variable. Here, it is 2, indicating a quadratic polynomial.
• Coefficients: These are constants multiplying the powers of the variable, like 2 in \( 2s \).
• Terms: Parts of the polynomial separated by '+', such as \( s^2 \) and \( 2s \).

Polynomials are crucial in various branches of mathematics, particularly in solving equations and modeling real-life situations. Mastering them helps students progress in algebra and calculus.