To solve the given problem, we are tasked with defining and interpreting the function [tex]\( h(x) \)[/tex]. The function is given by
[tex]\[ h(x) = (2x - 3)^5. \][/tex]
Here’s a detailed step-by-step explanation of the function:
1. Identify the Inner Expression:
- The inner expression of [tex]\( h(x) \)[/tex] is [tex]\( 2x - 3 \)[/tex].
2. Transformation by Outer Function:
- The function takes the inner expression [tex]\( 2x - 3 \)[/tex] and raises it to the power of 5.
3. Structure of the Function:
- The basic structure here is that any input [tex]\( x \)[/tex] will be fed into the linear expression [tex]\( 2x - 3 \)[/tex].
- The result will then be raised to the fifth power.
4. Understanding the Behavior for Specific Values of [tex]\( x \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = (2 \cdot 0 - 3)^5 = (-3)^5 = -243. \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = (2 \cdot 1 - 3)^5 = (-1)^5 = -1. \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ h(2) = (2 \cdot 2 - 3)^5 = (1)^5 = 1. \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ h(3) = (2 \cdot 3 - 3)^5 = (3)^5 = 243. \][/tex]
5. General Interpretation:
- The function [tex]\( h(x) = (2x - 3)^5 \)[/tex] is a polynomial function of degree 5.
- This function will exhibit the typical behavior of polynomials of odd degree, which includes having one end of the graph go to positive infinity and the other end to negative infinity, assuming we're considering the real number line.
In conclusion, the function defined by [tex]\( h(x) = (2x - 3)^5 \)[/tex] is a transformation where you input [tex]\( x \)[/tex], compute [tex]\( 2x - 3 \)[/tex], and then raise the result to the fifth power. The result from evaluating this function at various points helps us understand its general behavior and the types of values it will yield.
