Exploring the Functions f(x) 3x 2 and g(x) 5x a for Special a Values

Understanding the relationship between mathematical functions is a fundamental skill in algebra and calculus. In this article, we delve into the specific functions f(x) 3x 2 and g(x) 5x a. Our goal is to find the value of the constant a such that fg(x) gfx. This exploration will not only help in understanding these functions but also in utilizing algebraic methods to solve equations.

Introduction to the Functions

The functions f(x) and g(x) are linear functions, where f(x) 3x 2 and g(x) 5x a. We are tasked with finding the value of a that makes the composite functions equal. Specifically, we need to find when the composite function fg(x) is equal to gfx.

Calculating the Composite Functions

First, we calculate the composite functions fg(x) and gfx.

Calculating fg(x)

The composite function fg(x) is defined as g(f(x)). Given that f(x) 3x 2, we substitute x in g(x) with 3x 2.

So, we have:

fg(x)  g(3x   2)  5(3x   2)   a  15x   10   a  15x   (10   a)

Calculating gfx

The composite function gfx is defined as f(g(x)). Given that g(x) 5x a, we substitute x in f(x) with 5x a.

So, we have:

gfx  f(5x   a)  3(5x   a)   2  15x   3a   2  15x   (3a   2)

Setting the Functions Equal to Find a

Since fg(x) gfx, we can set the expressions for fg(x) and gfx equal to each other:

15x (10 a) 15x (3a 2)

By subtracting 15x from both sides, the equation simplifies to:

10 a 3a 2

Next, we solve for a by isolating it on one side of the equation. Subtract 2 from both sides:

8 a 3a

Subtract a from both sides:

8 2a

Divide both sides by 2 to find a:

a 4

Thus, the value of a that satisfies fg(x) gfx is 4.

Verification and Application

To verify, we can substitute a 4 back into the equations:

Verifying fg(x)

fg(x)  15x   (10   4)  15x   14

Verifying gfx

gfx  15x   (3(4)   2)  15x   14

As we can see, both fg(x) and gfx are indeed equal when a 4.

Conclusion

In conclusion, the value of a that makes the composite functions fg(x) and gfx equal is a 4. This problem demonstrates the importance of understanding how to manipulate and solve equations involving composite functions, which is a crucial skill in advanced algebra and calculus.