Problem 77 - Algebra and Trigonometry

Question

If \(h(x)\) is a function with range \([-5,8],\) what is the range of \(h(2 x+3) ?\)

Step-by-Step Solution

Answer

[-5, 8]

Step 1: Identify the range of the function

The given range of the function is \([-5, 8]\). This means that for any value of x, the output of the function \(h(x)\) will lie between -5 and 8, inclusive.

Step 2: Understand the transformation by the input function

The input to the function is transformed from \(x\) to \(2x + 3\). This transformation affects the domain of the function, but it does not change the range directly.

Step 3: Determine the range of the transformed function

Since we are only transforming the input variable and not the output, the range of \(h(2x + 3)\) remains the same as the original function \(h(x)\), which is \([-5, 8]\).

Key Concepts

function transformations

Function transformations involve changing the appearance or behavior of a function by applying specific operations, such as shifting, stretching, compressing, or reflecting. These transformations can be applied to the input (x-values) or the output (y-values) of the function.

For example, in the transformation from \(h(x)\) to \(h(2x + 3)\), we apply a linear transformation to the input. We multiply \(x\) by 2 and then add 3. This changes where the function is evaluated but not the actual output values.

Understanding how transformations affect a function can help us analyze and predict the behavior of complex functions. Always keep in mind that:

• Horizontal transformations involve changes to the x-values.
• Vertical transformations involve changes to the y-values.

range of a function

The range of a function refers to all possible output values (y-values) that the function can produce. For a function \(h(x)\) with a given range like \([-5, 8]\), it means that regardless of the input, the output will always lie within this interval.

When we talk about the range, we focus solely on the y-values. The range is crucial because it helps us understand the limitations and behavior of the function’s output. It answers the question: What y-values can this function produce?

• For instance, a function with the range \([-5, 8]\) will output values no lower than -5 and no higher than 8.
• This range remains the same even if the input to the function changes, as long as the output transformation (if any) is considered separately.

input transformations

Input transformations alter the way the x-values enter the function, which can significantly affect the function's graph but not necessarily its range. A clear example is transforming \(x\) to \(2x + 3\) in the function \(h(x)\).

This transformation impacts the input of the function without altering the possible output values directly. To understand this better, consider some key points about input transformations:

• They change the x-values used in the function but keep the y-values within the original range.
• In the case of the function \(h(2x + 3)\), although the function now responds to inputs that have been scaled and shifted, the outputs still lie within the range \([-5, 8]\).

Input transformations can make the function appear different on a graph, such as stretching or shifting the graph horizontally. However, the core range of the function remains intact, as the output values remain bound within the original interval.