Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

Let \(y=f(x)\) be a function with domain \(D=[-6,-2]\) and range \(R=[-10,-4]\). Find the domain \(D\) and range \(R\) for each function. (a) \(y=\frac{1}{2} f(x)\) (b) \(y=f(2 x)\) (c) \(y=f(x-2)+5\) (d) \(y=f(x+4)-1\) (e) \(y=f(-x)\) (f) \(y=-f(x)\) (g) \(y=f(|x|)\) (h) \(y=|f(x)|\)

Step-by-Step Solution

Answer

(a) D=[-6,-2], R=[-5,-2]; (b) D=[-3,-1], R=[-10,-4]; (c) D=[0,4], R=[-5,1]; (d) D=[-10,-6], R=[-11,-5]; (e) D=[-6,-2], R=[-10,-4]; (f) D=[-6,-2], R=[4,10]; (g) D=[-6,0], R=[-10,-4]; (h) D=[-6,-2], R=[4,10].

Step 1: Understand the given function and its domain and range

The original function is \(y = f(x)\) with domain \(D=[-6,-2]\) and range \(R=[-10,-4]\). All solution steps for the parts (a) to (h) will build on these domain and range values.

Step 2: Solve for part (a)

For the function \(y=\frac{1}{2} f(x)\), the domain remains unchanged as \(D' = [-6, -2]\) because this transformation only affects the output (range) by scaling. The range is scaled by \(\frac{1}{2}\), thus \(R' = \left[ \frac{-10}{2}, \frac{-4}{2} \right] = [-5, -2]\).

Step 3: Solve for part (b)

For the function \(y=f(2x)\), the input (domain) is compressed by a factor of 2. The original domain \([-6, -2]\) transforms into \([-3, -1]\) after dividing each value by 2. The range remains the same \(R' = [-10, -4]\).

Step 4: Solve for part (c)

For \(y=f(x-2)+5\), this translates the function 2 units to the right, adjusting the domain to \([0, 4]\). The range shifts up by 5 units, making \(R' = [-5, 1]\).

Step 5: Solve for part (d)

For \(y=f(x+4)-1\), the domain shifts left by 4 units becoming \([-10, -6]\) and the range shifts down by 1 unit, resulting in \(R' = [-11, -5]\).

Step 6: Solve for part (e)

\(y=f(-x)\) reflects the function over the y-axis. While the domain remains symmetric \([-2, 6]\), it is incorrect to reverse: it should simply reflect causing another description error here. Thus, the domain remains the same: \([-6, -2]\). The range remains \([-10, -4]\).

Step 7: Solve for part (f)

For \(y=-f(x)\), the reflection over the x-axis inverts the range: \(R' = [4, 10]\). The domain is unaffected: \([-6, -2]\).

Step 8: Solve for part (g)

For \(y=f(|x|)\), the symmetric property of absolute values manipulates the domain into non-negative inputs but keeps symmetry. Thus, domain extends to \([-6, 0]\) if \(f\) was defined for negatives, but advisory here aligns unchanged without new context. Range \(R' = [-10, -4]\) remains consistent.

Step 9: Solve for part (h)

For \(y=|f(x)|\), all negative range outputs turn positive. This returns the range: \(R' = [4, 10]\). The domain remains as \([-6, -2]\).

Key Concepts

Domain and Range

The domain and range of a function tell us about the possible input and output values. The domain is the set of input values that a function can take, while the range is the set of output values that a function can produce.
Understanding the domain and range is crucial in graphing functions as they help in determining how a function behaves within certain limits. For example:

• The original function has a domain \(D=[-6,-2]\) and a range \(R=[-10,-4]\).
• Different transformations affect either the domain or range or both, depending on how the function is altered.

For example, when you consider horizontal scaling or translation, primarily the domain is affected, whereas vertical transformations affect the range.

Function Notation

Function notation is a way to express functions in mathematics that gives clarity on inputs and outputs. For example, the function \(y = f(x)\) indicates that for any input \(x\), the function will provide an output \(y\).
This notation makes it easier to see the relationship between the input and output without explicitly writing all operations involved:

• The variable \(x\) represents input values which can change within the domain.
• The \(f\) in \(f(x)\) indicates that \(f\) is the function being applied to \(x\).

Using this notation provides a framework to explore function transformations easily, such as \(f(2x)\) showing horizontal scaling or \(-f(x)\) indicating a reflection over the x-axis.

Graph Transformations

Graph transformations take the entire graph of a function and modify it in some way. Key operations you might encounter include scaling, translating, and reflecting the graph.
Different types of transformations affect the graph:

• Vertical Scaling: Multiplying the function by a constant like \( \frac{1}{2} f(x)\) affects the range by compressing or stretching it.
• Horizontal Scaling: Modifying the input, such as \(f(2x)\), compresses the domain which affects the width of the graph.
• Reflection: Changing signs, as in \(-f(x)\) or \(f(-x)\), reflects the graph across the axes.
• Translations: Adding or subtracting numbers, for example \(f(x-2)+5\), shifts the graph horizontally and vertically.

By understanding how these transformations work, you can predict how the graph of a function will look after applying them.

Vertical and Horizontal Shifts

Vertical and horizontal shifts occur when you add or subtract a constant to a function or its variable. These shifts are a straightforward way to move the graph around, making them easy to apply and understand without needing to alter the shape of the graph.
Here's how these shifts work:

• Horizontal Shifts: These occur when you adjust the input \(x\) by adding or subtracting a constant. For instance, \(f(x-2)\) translates the graph 2 units to the right. Conversely, \(f(x+4)\) shifts it 4 units to the left.
• Vertical Shifts: Occur when you add or subtract from the function itself. For example, \(f(x)+5\) moves the graph up by 5 units, while \(f(x)-1\) moves it down 1 unit.

With these shifts, the key effect is on the placement of the graph, not on its shape. This makes them essential tools for graphing transformations.