f(x)=4x-1 g(x)=1/2x -1.dit(fog)(x)=?  (gof)(x)=?

Given the functions:

\[
f(x) = 4x - 1 \quad \text{and} \quad g(x) = \frac{1}{2}x - 1,
\]

we want to find the compositions \((f \circ g)(x)\) and \((g \circ f)(x)\).

### 1. Finding \((f \circ g)(x)\):

\((f \circ g)(x)\) means applying \(g(x)\) first and then applying \(f\). Mathematically, it can be written as:

\[
(f \circ g)(x) = f(g(x)).
\]

Substitute \(g(x)\) into \(f(x)\):

\[
f(g(x)) = f\left(\frac{1}{2}x - 1\right).
\]

Now, apply the definition of \(f(x)\):

\[
f\left(\frac{1}{2}x - 1\right) = 4\left(\frac{1}{2}x - 1\right) - 1.
\]

Simplify the expression:

\[
f\left(\frac{1}{2}x - 1\right) = 4 \cdot \frac{1}{2}x - 4 - 1 = 2x - 5.
\]

So, 

\[
(f \circ g)(x) = 2x - 5.
\]

### 2. Finding \((g \circ f)(x)\):

\((g \circ f)(x)\) means applying \(f(x)\) first and then applying \(g\). Mathematically, it can be written as:

\[
(g \circ f)(x) = g(f(x)).
\]

Substitute \(f(x)\) into \(g(x)\):

\[
g(f(x)) = g(4x - 1).
\]

Now, apply the definition of \(g(x)\):

\[
g(4x - 1) = \frac{1}{2}(4x - 1) - 1.
\]

Simplify the expression:

\[
g(4x - 1) = 2x - \frac{1}{2} - 1 = 2x - \frac{3}{2}.
\]

So,

\[
(g \circ f)(x) = 2x - \frac{3}{2}.
\]

### Final Answers:

\[
(f \circ g)(x) = 2x - 5 \quad \text{and} \quad (g \circ f)(x) = 2x - \frac{3}{2}.
\]

Would you like any further details or have any questions?

### Related Questions:

1. What are the domains of \((f \circ g)(x)\) and \((g \circ f)(x)\)?
2. How would you find the inverse of \((f \circ g)(x)\)?
3. What is the graphical interpretation of these compositions?
4. How would you determine the range of \((f \circ g)(x)\)?
5. How do these compositions change if you alter the constants in \(f(x)\) or \(g(x)\)?

**Tip:** When composing functions, always substitute one function directly into the other, carefully following order and simplifying step-by-step!

f(x) = 4x - 1, g(x) = 1/2x - 1. Find (f∘g)(x) and (g∘f)(x).

This detailed guide explores the function composition of f(x) = 4x - 1 and g(x) = 1/2x - 1. Learn step-by-step how to compute (f∘g)(x) and (g∘f)(x), including simplification techniques. Ideal for high school students studying function composition in algebra.

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