Composite Functions 53251A

1. **Problem Statement:** Find the composite functions $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$ for $f(x) = 2x - 1$ and $g(x) = x^2$. Also, state the domain of each composite function. 2. **Recall the definition of composite functions:** $$ (f \circ g)(x) = f(g(x)) \quad \text{and} \quad (g \circ f)(x) = g(f(x)) $$ 3. **Calculate each composite function:** (a) $f \circ g$: $$ (f \circ g)(x) = f(g(x)) = f(x^2) = 2(x^2) - 1 = 2x^2 - 1 $$ - Domain: Since $g(x) = x^2$ is defined for all real $x$, and $f$ is defined for all real numbers, the domain of $f \circ g$ is all real numbers, $\mathbb{R}$. (b) $g \circ f$: $$ (g \circ f)(x) = g(f(x)) = g(2x - 1) = (2x - 1)^2 $$ - Domain: $f(x)$ is defined for all real $x$, and $g$ is defined for all real numbers, so the domain of $g \circ f$ is all real numbers, $\mathbb{R}$. (c) $f \circ f$: $$ (f \circ f)(x) = f(f(x)) = f(2x - 1) = 2(2x - 1) - 1 = 4x - 2 - 1 = 4x - 3 $$ - Domain: Both $f$ functions are defined for all real $x$, so domain is $\mathbb{R}$. (d) $g \circ g$: $$ (g \circ g)(x) = g(g(x)) = g(x^2) = (x^2)^2 = x^4 $$ - Domain: $g$ is defined for all real $x$, so domain is $\mathbb{R}$. 4. **Summary:** - $f \circ g(x) = 2x^2 - 1$, domain: $\mathbb{R}$ - $g \circ f(x) = (2x - 1)^2$, domain: $\mathbb{R}$ - $f \circ f(x) = 4x - 3$, domain: $\mathbb{R}$ - $g \circ g(x) = x^4$, domain: $\mathbb{R}$