Problem 48 - Intermediate Algebra

Question

If \(f(x)=-2 x+7\), find \(f(a), f(a+2)\), and \(f(a+h)\).

Step-by-Step Solution

Answer

\(f(a) = -2a + 7, \ f(a+2) = -2a + 3, \ f(a+h) = -2a - 2h + 7\).

Step 1: Substitute for f(a)

First, we need to find what \(f(a)\) is by substituting \(a\) for \(x\) in the function \(f(x) = -2x + 7\). Substitute to get \(f(a) = -2a + 7\).

Step 2: Substitute for f(a+2)

Next, let's find \(f(a+2)\). Substitute \(a+2\) for \(x\) in the function: \(f(a+2) = -2(a+2) + 7\). Simplify it: \(f(a+2) = -2a - 4 + 7 = -2a + 3\).

Step 3: Substitute for f(a+h)

Now find \(f(a+h)\) by substituting \(a+h\) for \(x\): \(f(a+h) = -2(a+h) + 7\). Simplify it: \(f(a+h) = -2a - 2h + 7\).

Key Concepts

Function Evaluation

Function evaluation is about finding out what a function's output is when you plug a specific input into it. Think of a function like a machine: you insert an input (often called "x"), and the function does something to that input to produce an output. In mathematical terms, a function is usually represented as \( f(x) \), where \( f \) indicates the function and \( x \) is the input variable.

In this exercise, we started with the function \( f(x) = -2x + 7 \). To evaluate \( f(a) \), \( f(a+2) \), and \( f(a+h) \), we plug these different expressions for \( x \) into the function.

• For \( f(a) \), we substitute \( a \) for \( x \), resulting in \( f(a) = -2a + 7 \).
• For \( f(a+2) \), we replace \( x \) with \( a+2 \), leading to the result \( f(a+2) = -2a + 3 \).
• For \( f(a+h) \), we insert \( a+h \) in place of \( x \), giving us \( f(a+h) = -2a - 2h + 7 \).

By evaluating these different inputs, you see how the function changes based on what is plugged in.

Substitution Method

The substitution method involves replacing a variable within an expression with another value or expression. This method is very handy in algebra and is what allows us to find function evaluations.

In solving the original exercise, we used substitution to plug new expressions into the function \( f(x) = -2x + 7 \).

• In step one, we substituted \( x \) with \( a \) to get \( f(a) = -2a + 7 \).
• Next, \( x \) was replaced by \( a+2 \), which involved not only substituting but also simplifying the expression to obtain \( f(a+2) = -2a + 3 \).
• Lastly, the substitution \( x = a+h \) allowed us to find \( f(a+h) = -2a - 2h + 7 \).

Each of these steps required careful substitution, maintaining the integrity of the function setup. The key here is to correctly replace the variable \( x \) while accounting for every part of the expression being substituted.

Algebraic Expression Simplification

Simplifying algebraic expressions is a crucial skill in algebra, which involves reducing expressions to their simplest form. This not only makes expressions easier to work with but also helps in understanding the underlying patterns and relationships.

When simplifying, focus on combining like terms and reducing unnecessary complexity. Let's look at how this works in our problem:

• For \( f(a+2) = -2(a+2) + 7 \), start by distributing \(-2\): \( (-2) \cdot a \) and \( (-2) \cdot 2 \), leading to \(-2a - 4 + 7 \). Combine like terms to simplify to \(-2a + 3 \).
• With \( f(a+h) = -2(a+h) + 7 \), distribute \(-2\): \( -2a - 2h \). The full expression becomes \(-2a - 2h + 7\), which is already simplified as there are no like terms to combine.

Simplification requires attention to detail and practice to ensure each step reduces the expression appropriately. Remember, simplified expressions are often easier to interpret and use in further calculations.