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Function evaluation is a process where you plug a specific value into a function to find its output. In mathematical terms, it's like putting a number through a machine (the function) and seeing what comes out.
For instance, with the function \( g(t) = |t - 4| \), to evaluate \( g(3) \), we substitute \( t = 3 \) into the function. This gives you \( g(3) = |3 - 4| \). By calculating \( 3 - 4 \), we find \( -1 \), and then we transform this into its absolute value, which is \( 1 \).
Therefore, function evaluation helps us understand what output we get for a specific input value in the function equation or expression.

The concept of absolute value is closely related to the idea of distance on a number line. Absolute value is a number's distance from zero. This means we don't care about the direction; we just want to know how far a number is from zero.
In our example, \( g(t) = |t - 4| \) tells us the distance of \( t \) from \( 4 \). When evaluating \( g(3) \), we find that \( 3 - 4 = -1 \), which is \( 1 \) unit away from zero after taking the absolute value.
Thus, the function essentially measures how far a number \( t \) is from \( 4 \) on the number line. Whether we move right or left, the absolute value gives us a positive distance to evaluate. It's a handy way to always find non-negative distance values.

Substitution is a method for replacing variables in a function with numbers or expressions to find a new function or value. To do so, identify the variable to replace, substitute it with the new number or expression, and simplify accordingly.
For example, to evaluate \( g(x+4) \) in the function \( g(t) = |t-4| \), we replace \( t \) with \( x+4 \). This gives \( g(x+4) = |(x+4) - 4| \).
By simplifying, \( (x+4) - 4 = x \), we see that \( g(x+4) = |x| \). This substitution process allows us to understand how the function behaves when the input variable is altered, providing insight into more generalized function outcomes beyond just numbers.