Composition and Inverse Functions in Algebra

Function Difference and Operations

This paper works through the assigned problems from week five of an Intermediate Algebra course. The initial task is to calculate (f − h)(4) using the defined functions f(x) = 2x + 5, g(x) = x² − 3, and h(x) = 7 − x³.

To calculate (f − h)(4), we first evaluate f(4) and h(4) separately, then subtract the results.

For f(4):
f(4) = 2(4) + 5 = 13
This is computed by multiplying 2 times 4 and then adding 5.

For h(4):
h(4) = 7 − 4³ = 7 − 64 = −57
We calculate 4³, then subtract from 7.

Function Composition

The final result is:
(f − h)(4) = 13 − (−57) = 70

Function composition combines two or more functions by substituting one function into another. We evaluate both (f ∘ g)(x) and (h ∘ g)(x).

(f ∘ g)(x):
We substitute g(x) into f(x):
f(g(x)) = 2(x² − 3) + 5
= 2x² − 6 + 5
= 2x² − 1

Graphing and Transformation

(h ∘ g)(x):
We substitute g(x) into h(x):
h(g(x)) = 7 − (x² − 3)³
This expression represents the composition of h with g, where the output of g becomes the input to h.

The original function g(x) = x² − 3 is a parabola with vertex at (0, −3). To shift the graph 6 units to the right and 7 units down, we apply horizontal and vertical transformations.

A horizontal shift of 6 units to the right replaces x with (x − 6).
A vertical shift of 7 units down subtracts 7 from the function value.
The transformed function is: g(x) = (x − 6)² − 3 − 7 = (x − 6)² − 10

Inverse Functions

When expanded, this becomes:
g(x) = x² − 12x + 36 − 10 = x² − 12x + 26
The vertex of the transformed parabola is at (6, −10).

Inverse functions reverse the operation of the original function. For f(x) = 2x + 5, we find f⁻¹(x) by swapping variables and solving for y.

Finding f⁻¹(x):
Start with y = 2x + 5
Swap x and y: x = 2y + 5
Solve for y:
x − 5 = 2y
y = (x − 5)/2
Therefore, f⁻¹(x) = (x − 5)/2

Finding h⁻¹(x):
Start with y = 7 − x³
Swap x and y: x = 7 − y³
Solve for y:
y³ = 7 − x
y = ∛(7 − x)
Therefore, h⁻¹(x) = ∛(7 − x)

Conclusion

To verify an inverse function, we can check that f(f⁻¹(x)) = x. Understanding inverse functions is essential for solving equations and modeling real-world relationships where we need to reverse a process.

This paper has demonstrated mastery of key intermediate algebra concepts: function operations, composition, transformation, and inverse functions. Over the course of ten weeks, these topics have been studied systematically, supported by resources such as the ALEKS platform, which provided interactive practice and immediate feedback. The combination of algebraic theory and computational practice has developed both conceptual understanding and procedural fluency. Completing this unit represents significant progress toward proficiency in algebraic reasoning and prepares for advanced mathematics coursework.