Problem 37 - Beginning and Intermediate Algebra

Question

Let \(f(x)=3 x-7\) and \(g(x)=x^{2}-4 x-9 .\) Find each of the following and simplify. $$g(t+2)$$

Step-by-Step Solution

Answer

\(g(t + 2) = t^2 - 3\)

Step 1: Substitute t + 2 for x in g(x)

We will start by replacing x in the function g(x) with t + 2: \(g(t + 2) = (t + 2)^2 - 4(t + 2) - 9\)

Step 2: Expand the expression

Now, expand the expression by squaring (t + 2) and distributing -4 into (t + 2): \(g(t + 2) = (t^2 + 4t + 4) - 4t - 8 - 9\)

Step 3: Simplify the expression

Now we'll combine like terms to simplify the expression: \(g(t + 2) = t^2 + 4t - 4t + 4 - 8 - 9\) \(g(t + 2) = t^2 - 3\) So, the simplified expression for \(g(t + 2)\) is: \(g(t + 2) = t^2 - 3\)

Key Concepts

Understanding Function Substitution

Function substitution is a method where you replace a variable in a function with another expression.
It allows you to evaluate the function for different inputs. This is an essential skill in algebra functions.

In the given exercise, you substitute the expression \( t + 2 \) into the function \( g(x) = x^2 - 4x - 9 \). Here's how function substitution works:

• Identify the function and the expression you need to substitute. In our example, it's \( g(x) \) and \( t + 2 \).
• Replace every occurrence of \( x \) in \( g(x) \) with \( t + 2 \). This leads to \( g(t + 2) = (t+2)^2 - 4(t+2) - 9 \).
• Once substituted, the function is ready for further simplification or evaluation.

By practicing this technique, you improve your problem-solving skills in algebra and build a solid foundation for more advanced topics.

Mastering Polynomial Simplification

After substitution, we need to simplify the resulting polynomial. Polynomial simplification involves expanding and reducing expressions to their simplest form.
It makes the results easier to understand and further work with.

In this exercise, after substituting, we have:\[(t + 2)^2 - 4(t + 2) - 9\]To simplify:

• Expand \((t + 2)^2\) to \(t^2 + 4t + 4\).
• Distribute \(-4\) across \(t + 2\), resulting in \(-4t - 8\).
• Rewrite and combine like terms: \(t^2 + 4t + 4 - 4t - 8 - 9\).

This can then be simplified to \(t^2 - 3\), by canceling terms and simply adding or subtracting numbers. Simplifying polynomials helps in making functions easier to visualize and analyze.

Introduction to Quadratic Expressions

Quadratic expressions are algebraic expressions that involve terms up to the second degree or square. They are significant because they form the basis of quadratic equations and functions.
Quadratics have a unique property; they can be graphed as a parabola—inverted or upright depending on the leading coefficient.

The expression \( t^2 - 3 \) is a simplified quadratic expression from our exercise. It includes:

• A leading term \( t^2 \), indicating it is a quadratic.
• A constant \(-3\), which shifts the parabola vertically on the graph.

Recognizing and working with quadratic expressions prepares you for solving quadratic equations and understanding graphs in algebra. You'll find quadratics appear frequently in various contexts, such as physics, engineering, and beyond.