Problem 76 - A Graphical Approach to College Algebra

Question

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$\frac{1}{3} f(x-3)$$

Step-by-Step Solution

Answer

Domain: [2, 5]; Range: [0, 1].

Step 1 - Understand the given function

We have a function \(f(x)\) with a specified domain of \([-1, 2]\) and range \([0, 3]\). Our task is to find the domain and range of the transformed function \(\frac{1}{3} f(x-3)\).

Step 2 - Analyze the transformation on the domain

The expression \(f(x-3)\) shifts the graph of \(f(x)\) to the right by 3 units. Original domain \([-1, 2]\) is transformed to \([2, 5]\) by adding 3 to each endpoint.

Step 3 - Determine the domain of the transformed function

Since the transformation involves shifting the function to the right by 3 units, the new domain of \(\frac{1}{3} f(x-3)\) is \([2, 5]\).

Step 4 - Analyze the transformation on the range

The function \(\frac{1}{3} f(x-3)\) involves multiplying the output of \(f(x-3)\) by \(\frac{1}{3}\). This scales the original range \([0, 3]\) by \(\frac{1}{3}\), resulting in the range \([0, 1]\).

Step 5 - Determine the range of the transformed function

The multiplication by \(\frac{1}{3}\) scales every value in the original range, so the new range is \([0, 1]\).

Key Concepts

Domain

In mathematics, the domain of a function refers to all the possible input values (usually represented by \(x\) values) that a function can accept without resulting in undefined behavior. For the function \(f(x)\), the given domain is \([-1, 2]\). This means the function is defined and can accept any \(x\) values from -1 to 2 inclusively.

Now, to find the domain of the transformed function \(\frac{1}{3} f(x-3)\), we need to consider the impact of the transformation \(x - 3\). This expression indicates a horizontal shift. Specifically, it moves the entire graph of \(f(x)\) right by 3 units.

• Add 3 to each endpoint of the original domain.
• Transform \([-1, 2]\) to \([2, 5]\).

Thus, the domain of the new function \(\frac{1}{3} f(x-3)\) becomes \([2, 5]\). Each value in this range is valid for the new transformed function.

Range

The range of a function represents all possible output values (often represented by \(y\) values) that a function can produce. For the given function \(f(x)\), the range is \([0, 3]\), meaning that the possible outputs for this function are values between 0 and 3, inclusively.

When we examine the transformation \(\frac{1}{3} f(x-3)\), it involves multiplying each output of \(f(x-3)\) by \(\frac{1}{3}\).

• This operation scales the original range by \(\frac{1}{3}\).
• The original outputs that ranged from 0 to 3 are now transformed to a new range of \([0, 1]\).

Therefore, the range of the transformed function \(\frac{1}{3} f(x-3)\) is \([0, 1]\), reflecting the impact of the scaling factor on the function's outputs.

Function Shifting

Function shifting is a transformation that moves the graph of a function either horizontally or vertically without changing its shape. In our problem, \(f(x)\) undergoes a horizontal shift represented by \(f(x-3)\).

This specific transformation causes the entire graph of \(f(x)\) to move 3 units to the right.

• Horizontal shifts are achieved by adding or subtracting a constant from \(x\) in the function.
• Adding \(-3\) to \(x\) shifts the graph right, impacting the original domain \([-1, 2]\) to become \([2, 5]\).

The effect on the graph looks as if you've grabbed it and moved it to the side along the \(x\)-axis, making all points move to the right while maintaining their relative positions.

Scaling Function

A scaling of a function involves expanding or compressing the graph in the vertical direction. This transformation directly affects the output values (range) of the function.

In the expression \(\frac{1}{3} f(x-3)\), the function's output values are scaled by the factor of \(\frac{1}{3}\).

• This means that every \(y\)-value is reduced to one-third of its original value.
• The range of \(f(x)\) was \([0, 3]\), but once each output is multiplied by \(\frac{1}{3}\), the range becomes \([0, 1]\).

Each original point on the curve is thus vertically pulled closer to the \(x\)-axis, compressing the range while preserving the relative layout of points.