Composite Functions 04053F

1. **State the problem:** We are given two functions: $$f(x) = 7x + 8$$ $$g(x) = 4x$$ We need to find the composite functions: (a) $f \circ g$ (which means $f(g(x))$) (b) $g \circ f$ (which means $g(f(x))$) (c) $f \circ f$ (which means $f(f(x))$) (d) $g \circ g$ (which means $g(g(x))$) and state the domain of each. --- 2. **Recall the rule for composite functions:** The composite function $f \circ g$ is defined as $f(g(x))$, which means you substitute $g(x)$ into $f$. Since both $f$ and $g$ are linear functions defined for all real numbers, their domains are all real numbers. --- 3. **Calculate each composite function:** (a) $f \circ g = f(g(x)) = f(4x)$ $$f(4x) = 7(4x) + 8 = 28x + 8$$ Domain: Since $g(x)$ is defined for all real $x$ and $f$ is defined for all real inputs, the domain of $f \circ g$ is all real numbers. (b) $g \circ f = g(f(x)) = g(7x + 8)$ $$g(7x + 8) = 4(7x + 8) = 28x + 32$$ Domain: Both $f$ and $g$ are defined for all real numbers, so domain is all real numbers. (c) $f \circ f = f(f(x)) = f(7x + 8)$ $$f(7x + 8) = 7(7x + 8) + 8 = 49x + 56 + 8 = 49x + 64$$ Domain: All real numbers. (d) $g \circ g = g(g(x)) = g(4x)$ $$g(4x) = 4(4x) = 16x$$ Domain: All real numbers. --- 4. **Summary:** - $f \circ g(x) = 28x + 8$, domain: all real numbers - $g \circ f(x) = 28x + 32$, domain: all real numbers - $f \circ f(x) = 49x + 64$, domain: all real numbers - $g \circ g(x) = 16x$, domain: all real numbers Each composite function is a linear function defined for all real numbers.