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Function substitution is a crucial concept in algebra. It involves replacing the variable in a function with a specific value or another expression. For example, if you have a function \(f(x) = -2x + 3\), and you want to evaluate \(f(3)\), you substitute 3 for every \(x\) in the function:

\(f(3) = -2(3) + 3 = -6 + 3 = -3\).

This process allows you to find the value of the function at a particular point. It's like a recipe: you follow steps to substitute the given number and simplify the equations.

Keep in mind that this concept is not only for numbers but can be extended to algebraic expressions as well. For instance, evaluating \(f(a+1)\) would involve substituting \(a+1\) for \(x\) in the function \(f(x)\).

Dividing functions involves applying the basic principles of division from arithmetic to functions. If you have two functions, say \(f(x)\) and \(g(x)\), and you need to find \(\frac{f(3)}{g(3)}\), you first evaluate each function individually at that point and then perform the division.

In the example provided:\(f(x) = -2x + 3\) and \(g(x) = x^2 - 5\).
First, we find \(f(3) = -3\) and \(g(3) = 4\), then we compute:
\(\frac{f(3)}{g(3)} = \frac{-3}{4}\).

This concept helps in understanding more complex function operations and prepares you for calculus, where you will often divide functions to find derivatives and integrals.

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. Simplifying these expressions involves reducing them to their simplest form without changing their value.

Consider the expression \(-2x + 3\). To simplify:\

• Apply the distributive property if necessary.
• Combine like terms.
• Substitute any given values and calculate.

In evaluating \(f(x) = -2x + 3\) at \(x = 3\), you substitute and simplify:
\(f(3) = -2(3) + 3 = -3\).

This helps ensure that you correctly interpret and manage expressions in various mathematical contexts, such as solving equations and performing operations with functions.