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Function substitution is the practice of replacing the variable in a function with another value or expression. In the context of the absolute value function, this involves replacing the variable 'x' in our function, \(f(x) = |x| + 4\), with the given values or expressions. This is a way of evaluating how the function behaves when different values are inputted. * To find \(f(a)\), we substitute \(a\) in place of \(x\) in the function, resulting in \(f(a) = |a| + 4\).* Similarly, to find \(f(b+1)\), substitute \(b+1\) into the function, transforming it to \(f(b+1) = |b+1| + 4\).* For \(f(3x)\), replace \(x\) with \(3x\) to obtain \(f(3x) = |3x| + 4\).These substitutions help us determine the function's output for specific inputs, demonstrating how versatile the function can be when applied to various scenarios.

Mathematical operations are processes we perform to manipulate numbers or expressions. In the context of substitution into an absolute value function, two main operations are involved: taking an absolute value and performing addition. Taking the absolute value, indicated by the bars \(| \cdot |\), means determining the non-negative distance of a number from zero, regardless of its original sign. * For \(f(a)\), \(|a|\) means if \(a\) is negative, we change it to a positive. If \(a\) is positive, it stays the same. * Similarly, \(|b+1|\) and \(|3x|\) involve evaluating each expression within the absolute value bars to ensure the output is non-negative. After calculating the absolute value, the next step is to add 4. This constant addition shifts the entire function output by 4 units upwards on the graph. Each step of substitution and operation is fundamental to understanding how different expressions affect the output of the function \(f(x) = |x| + 4\).

Evaluating functions refers to the process of finding the output of a function for different inputs. With absolute value functions like \(f(x) = |x| + 4\), evaluating involves substituting particular values or expressions into \(x\) and then carrying out the specified operations. To efficiently evaluate:* Find the expression or value that will replace \(x\).* Perform the mathematical operations within the function. Take the absolute value of the substituted expression, then add the constant value (4 in this case). For example, after substituting \(a\) in \(f(a)\), you calculate \(|a|\) and then proceed to add 4, giving \(f(a) = |a| + 4\). This step-by-step process ensures you correctly determine the function's value at various points and understand how the function behaves under different substitutions.