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In function composition, substitution is a vital concept. To find a specific value, we substitute a given expression into the function. For example, consider the functions provided in the exercise: f(x) = 3x + 1 and g(x) = x^2 - 2. To find \( f(a + h) \), we substitute \(a + h\) into \(f(x)\).Substitution works like replacing variables in algebra. When we substitute \(a + h\) for \(x\) in function \(f(x)\), every occurrence of \(x\) in \(3x + 1\) becomes \(a + h\). This gives us: \[f(a + h) = 3(a + h) + 1 \]. Understanding substitution helps simplify complex expressions and find values more easily.

Algebraic expressions are the combination of variables, constants, and mathematical operations. In our problem, the function \(f(x) = 3x + 1\) is an algebraic expression, as it consists of the variable \(x\), a constant (1), and mathematical operations (multiplication and addition).When we substitute an expression like \(a + h\) into \(f(x)\), we modify the algebraic expression. This is an essential skill in algebra because many problems require altering expressions to discover a solution.Consider the new expression obtained after substitution: \[f(a + h) = 3(a + h) + 1 \]. This expression can still be simplified. Algebraic manipulation enables us to rewrite and understand expressions in different forms.

Simplifying equations involves combining like terms and performing arithmetic operations to present the expression more clearly. After substitution in our exercise, we got: \[f(a+h) = 3(a+h) + 1\]. We now simplify this expression:* Distribute the 3: \[3(a + h)\] becomes \[3a + 3h\]. * Add the constant 1: So, \[f(a+h) = 3a + 3h + 1\]. Simplifying equations helps us understand the relationships between variables better. It makes calculations easier and aids in solving more complex problems.