Problem 58 - Algebra 2

Question

Let \(g(x)=3 x+2\) and \(f(x)=\frac{x-2}{3} .\) Find each value. $$ f(g(1)) $$

Step-by-Step Solution

Answer

Therefore, the value of \(f(g(1))\) is 1.

Step 1: Find the Value of g(1)

Plug in 1 in place of x in the function g(x) so that it becomes \(g(1)=3(1)+2\). Solve this equation and it becomes \(g(1)=5\).

Step 2: Find the Value of f(g(1))

Now, we've gotten the output from function g(1) to be 5. This output will be used as an input in function f(x), hence it will be substituted in f(x) as \(f(5)\). So use this value to find \(f(5)\) as \(f(5)=\frac{5-2}{3}\). Calculate this to get \(f(5)=1\).

Key Concepts

Linear Functions

A linear function is a type of function that creates a straight line when it's graphed. Linear functions are represented in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the function \(g(x) = 3x + 2\), the slope \(m\) is 3, which indicates that for every step we move along the x-axis, the function's value increases by 3. The y-intercept \(b\) is 2, meaning the line crosses the y-axis at the point (0, 2).
Understanding linear functions is crucial because they model relationships where change is constant. In real-world scenarios, linear models are used to describe anything from the distance traveled by a car moving at a constant speed to the price of goods increasing at a fixed rate.

• Slope: Determines the steepness and direction of the line.
• Y-intercept: Gives the starting value of the function when x is zero.

Recognizing these parts of a linear function helps immensely in graphing and predicting trends based on the given formula.

Function Evaluation

Function evaluation involves finding outputs (or results) of a function for specific inputs. This is done by substituting the input value into the function equation. Let's look at the function \(g(x) = 3x + 2\). To evaluate this function for an input of 1, you replace \(x\) with 1 in the equation to find \(g(1)\).

Performing these steps is straightforward:

• Identify the function, such as \(g(x)\).
• Substitute the input value for \(x\) in the function.
• Simplify the equation to find the answer.

In our example, this meant calculating \(g(1) = 3(1) + 2\), which simplifies to 5. Thus, \(g(1)\) equals 5. Function evaluation is a foundational skill in calculus and further math studies, leading to understanding more complex ideas like derivatives and integrals.

Inverse Functions

Inverse functions essentially undo what the original function does. If you have a function \(f(x)\) and its inverse function \(f^{-1}(x)\), applying \(f\) and then \(f^{-1}\) to any input will return you to the original value. This concept is particularly useful in solving equations and modeling real-world situations where you need to reverse a process.

The function \(f(x) = \frac{x-2}{3}\) serves as the inverse of \(g(x) = 3x + 2\). Here's why:

• When you use \(g\) on an input and then \(f\) on the result, you return to the original input. In our problem, we used \(g(1)\) to find 5, and \(f(5)\) brought us back to 1.
• Inverse functions have a specific algebraic property: \(f(g(x)) = x\).

Understanding inverse functions allows you to tackle problems where finding the original input is necessary, such as decrypting encoded data or converting temperatures between Celsius and Fahrenheit.