Problem 47 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. $$g(5 t)$$

Step-by-Step Solution

Answer

The short answer for the given question is: \(g(5t) = 25t^2 + 35t + 2\)

Step 1: Identify the g(x) function

We are given the function g(x) as follows: $$ g(x) = x^2 + 7x + 2 $$

Step 2: Substitute 5t for x in the g(x) function

Now, we need to replace x with 5t in the g(x) function: $$ g(5t) = (5t)^2 + 7(5t) + 2 $$

Step 3: Simplify the function g(5t)

Next, we simplify the terms: $$ g(5t) = 25t^2 + 35t + 2 $$ So, the simplified expression for g(5t) is: $$ g(5t) = 25t^2 + 35t + 2 $$

Key Concepts

Polynomial Functions

Polynomial functions are expressions that may include variables raised to a non-negative integer power, coefficients, and constants. In general, a polynomial in one variable can be expressed in the form:

• Coefficient times variable raised to a power (a_ix^ia), where a_ia is a constant.
• A constant term, often written as the last component of the polynomial.

For example, \(g(x) = x^2 + 7x + 2\) is a polynomial function. It includes terms like \(x^2\) and \(7x\), with coefficients 1 and 7, respectively, and a constant 2. Polynomial functions can vary in degree, which is determined by the highest power of the variable. In our example, the degree is 2, which indicates the highest power is \(x^2\). Polynomial functions can model a wide variety of real-world scenarios, which makes them important in algebra.

Substitution in Functions

Substitution is a key operation when working with functions; it involves replacing the variable with a specific value or another expression. This technique helps find particular outputs or further simplify the function, changing its form according to needs.
When solving \(g(x) = x^2 + 7x + 2\)and we need to evaluate \(g(5t)\), substitution allows us to redefine the function using the new input \(5t\). In practice, replace all occurrences of \(x\) with \(5t\) resulting in: \(g(5t) = (5t)^2 + 7(5t) + 2\).
Remember, substitution helps transform the function by plugging in the chosen variable or number seamlessly, ensuring accurate evaluations or further simplifications.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more concise or standard form. It means combining like terms, applying arithmetic operations, and following the order of operations to give an expression its simplest version.
In the example of \(g(5t) = (5t)^2 + 7(5t) + 2\), we simplify:

• Calculate \((5t)^2\), which results in \(25t^2\).
• Next, evaluate \(7(5t)\) resulting in \(35t\).
• Finally, include the constant \(+ 2\) without changes.

Thus, the expression simplifies to \(25t^2 + 35t + 2\). Simplifying makes the expression easier to interpret and use in further calculations.

Quadratic Functions

Quadratic functions are a specific type of polynomial function where the degree is exactly 2. The standard form of a quadratic function is \(ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants and \(a eq 0\).
The example \(g(x) = x^2 + 7x + 2\) is a quadratic function as its highest degree term is \(x^2\).
Quadratic functions are significant due to their parabolic shape when graphed, displaying either upward or downward opening, determined by the sign of \(a\).

• They have optimal properties like finding maximum or minimum points.
• They often appear in physical contexts like projectile motion or in calculating areas.

Understanding quadratic functions is crucial for solving various algebraic problems and interpreting their graphical cases in real-world applications.