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Substitution is a key concept in mathematics, particularly when dealing with functions. It involves replacing a variable with a specific value. In the context of our exercise, we are given functions and need to find their values at a specific point. For example, we substitute \(-2\) into the function \(f(x) = -2x + 3\). This means wherever we see \(x\), we replace it with \(-2\). Therefore, \(f(-2) = -2(-2) + 3\). This substitution simplifies to \(4 + 3 = 7\). Substitution helps us evaluate functions at specific points and is often the first step in solving function-related problems.

Algebraic functions are mathematical expressions involving variables and constants combined using algebraic operations such as addition, subtraction, multiplication, division, and exponentiation. In our example, \(f(x) = -2x + 3\) and \(g(x) = x^2 - 5\) are algebraic functions.
These functions can be evaluated by substituting specific values for their variables and then simplifying the expression using basic algebraic rules. For instance, substituting \(-2\) into \(-2x + 3\) transforms it into \(-2(-2) + 3\), which simplifies to \(4 + 3 = 7\). Similarly, substituting \(-2\) into \(x^2 - 5\) transforms it into \((-2)^2 - 5\), simplifying to \(4 - 5 = -1\). Understanding and manipulating algebraic functions is crucial for solving various mathematical problems.

Function multiplication involves multiplying the values of two functions evaluated at the same point. In the given exercise, we found \(f(-2) = 7\) and \(g(-2) = -1\). To find \(f(-2) \cdot g(-2)\), we multiply these two results together: \(7 \cdot -1 = -7\). This process is essential in many applications, such as calculating combined effects or solving complex equations.
When performing function multiplication, always ensure that both functions are correctly evaluated at the same input value before multiplying their results. This technique is useful not only in algebra but also in calculus and other advanced mathematical fields.