Problem 9 - Algebra for College Students

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 the variable for f(a)

To find \( f(a) \), we substitute \( a \) into the function \( f(x) = -2x + 7 \). So, \( f(a) = -2a + 7 \).

Step 2: Substitute the expression for f(a+2)

To find \( f(a+2) \), we substitute \( a+2 \) into the function \( f(x) = -2x + 7 \). Thus, \( f(a+2) = -2(a+2) + 7 \). Simplify this expression to get \( f(a+2) = -2a - 4 + 7 = -2a + 3 \).

Step 3: Substitute the expression for f(a+h)

To compute \( f(a+h) \), substitute \( a+h \) into the function \( f(x) = -2x + 7 \). The expression becomes \( f(a+h) = -2(a+h) + 7 \). Simplify it to obtain \( f(a+h) = -2a - 2h + 7 \).

Key Concepts

Substitution Method

The substitution method is a key concept when it comes to evaluating functions. When you have a function like \( f(x) = -2x + 7 \), you're essentially given a machine that turns input values into output values according to a specific rule.
To evaluate \( f(a) \), \( f(a+2) \), and \( f(a+h) \), the substitution method involves the direct replacement of \( x \) in the given function with the desired input.

• For \( f(a) \), every occurrence of \( x \) in the function is replaced with \( a \), yielding \( f(a) = -2a + 7 \).
• Similarly, for \( f(a+2) \), replace \( x \) with \( a+2 \), resulting in \( f(a+2) = -2(a+2) + 7 \). Further simplification shows \( f(a+2) = -2a + 3 \).
• For \( f(a+h) \), substitute \( x \) with \( a+h \) to get \( f(a+h) = -2(a+h) + 7 \), which simplifies to \( f(a+h) = -2a - 2h + 7 \).

The substitution method is helpful because it is a very systematic way of finding the value of a function at different points.
By changing the input variable, you can explore how the function behaves with different inputs.

Linear Functions

Linear functions are among the simplest and most important types of functions in algebra. They have the general form \( f(x) = mx + b \).
In our exercise, the linear function is \( f(x) = -2x + 7 \). Here, the coefficient \( -2 \) is known as the slope, and \( 7 \) is the y-intercept. The slope indicates the steepness and direction of the line associated with the function.

• A positive slope means the line goes upwards as it moves from left to right.
• A negative slope, like \( -2 \), means the line goes downwards.
• In this case, the function decreases by 2 units for every 1 unit increase in \( x \).

Understanding linear functions is crucial since they appear frequently in many areas of math and real-world situations, such as calculating speed or predicting costs.
They provide a straightforward model for how one quantity affects another.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operations. They are essential for solving many mathematical problems.
In our function \( f(x) = -2x + 7 \), \(-2x + 7\) is an algebraic expression. The terms are \( -2x \) and \( 7 \).
Substituting variables or expressions into algebraic expressions is a fundamental skill. It involves replacing a variable with another expression.

• When substituting \( a \) for \( x \), the expression becomes \( -2a + 7 \).
• Substituting \( a+2 \) gives \( -2(a+2) + 7 \), which simplifies to \( -2a + 3 \).
• For \( a+h \), the expression \( -2(a+h) + 7 \) simplifies to \( -2a - 2h + 7 \).

Algebraic expressions are versatile as they allow flexibility in representing a wide range of problems and scenarios.
Understanding how to manipulate these expressions is crucial for problem-solving.