Problem 28 - College Algebra Essentials

Question

Evaluate each function at the given values of the independent variable and simplify. $$ f(x)=3 x+7 $$ a. \(f(4)\) b. \(f(x+1) \) c. \(f(-x)\)

Step-by-Step Solution

Answer

The values of the function at the given points are: a. \( f(4) = 19 \) b. \( f(x + 1) = 3x + 10 \) c. \( f(-x) = -3x + 7 \)

Step 1: Evaluate \( f(4) \)

Substitute \( x = 4 \) into the function equation. This gives \( f(4) = 3*4 + 7 \). After simplifying this, it equals 19.

Step 2: Evaluate \( f(x + 1) \)

Next, substitute \( x = x + 1 \) into the function equation. This results in \( f(x + 1) = 3*(x + 1) + 7 \). To simplify this, distribute the 3 through the parentheses to get \( 3x + 3 + 7 = 3x + 10 \).

Step 3: Evaluate \( f(-x) \)

Finally, substitute \( x = -x \) into the function equation. This results in \( f(-x) = 3*(-x) + 7 \). To simplify, multiply the 3 and \( -x \) to get \( -3x + 7 \).

Key Concepts

Substitution in Functions

When we talk about substitution in functions, we are referring to the process of replacing the variable (usually "x") in the function equation with a specific number or expression. This process is essential in evaluating the function for different values.

Let's consider our exercise, where we have the function \(f(x) = 3x + 7\). Here, we are asked to evaluate the function at different values, such as \(f(4)\), \(f(x+1)\), and \(f(-x)\). Each of these represents a different substitution of "x" with a particular value or expression.

• For \(f(4)\), substitute "x" with 4.
• For \(f(x+1)\), substitute "x" with \(x+1\).
• For \(f(-x)\), substitute "x" with \(-x\).

It's as simple as replacing throughout the function where you see "x" with the given value, allowing us to evaluate the function under new conditions.

Simplifying Expressions

Once we substitute a value into a function, the next step is to simplify the resulting expression. Simplifying involves performing the operations in the equation to get to the simplest form.

In our exercise, after substituting we simplify the resulting expressions. Let's break it down:

• For \(f(4) = 3*4 + 7\), calculating \(3 \times 4\) gives us 12, then adding 7 gives a final result of 19.
• For \(f(x+1) = 3(x+1) + 7\), start by using the distributive property: \(3 \times (x+1)\) becomes \(3x + 3\). Then add 7 to get the final simplified form \(3x + 10\).
• For \(f(-x) = 3(-x) + 7\), multiplying 3 by \(-x\) results in \(-3x\), and adding 7 gives the simplified expression \(-3x + 7\).

Simplifying not only helps us find clearer and more accurate results but also makes complex ideas much easier to understand.

Linear Functions

A linear function is a type of mathematical function that creates a straight line when graphed. It commonly takes the form \(f(x) = ax + b\), where "a" and "b" are constants.

The function in our exercise, \(f(x) = 3x + 7\), is a linear function because it fits this form.

• The "a" value, 3, indicates the slope of the line, showing how steep the line is.
• The "b" value, 7, is the y-intercept, telling us where the line crosses the y-axis.

Linear functions are fundamental because they represent a constant rate of change, making them easy to work with for predictions and finding values through simple substitutions and arithmetic operations. Understanding this function's behavior helps in seeing how each substitution transforms the equation while maintaining its linear characteristics.