Values of functions are numbers. So, it is no surprise that we can make new functions
by combining old functions with addition, subtraction, multiplication, and division.
Thinking ahead to Calculus, it is also very handy to see multiplication by a
constant.
When categorizing a formula, our first thought is that the function is an elementary
function and we would categorize it from our library.
If it is not, then we turn to the structure of the expression where the Order of
Operations dictates how we view the algebra.
When thinking of the Order of Operations, we are thinking of what the expression actually looks like as it is written.
These are “Is” questions.
We are not thinking of how the expression might be changed algebraically. We are
identifying how it is currently written.
The Order of Operations instructs us on how to make this decision.
Constant Multiple
If \(f\) is a function and \(A\) is a constant, then \(A \, f\) is called a constant multiple of \(f\).
The domain of a constant multiple of a function is equal to the domain of the
function.
If \(f\) belongs to a nice category of functions, then the constant multiple also belongs to the same nice category.
\(f(x) = 4 \, e^x\) is a constant multiple of \(e^x\).
\(Y(t) = -5 \, \sin (4t + 8)\) is a constant multiple of \(\sin (4t + 8)\).
\(m(k) = 4 (k+1)(k-5)\) is a constant multiple of \((k+1)(k-5)\).
\(T(n) = 4 (n+1) + 6\) is not a constant multiple of \((n+1) + 6\).
Sums
If \(f\) and \(g\) are functions, then \(f + g\) is called the sum of \(f\) and \(g\).
The sum of two functions is defined on the intersection of their two domains.
\(f(x) = e^x\) is an exponential function.
\(g(t) = 3 t + 4\) is a linear function.
However, \((f + g)(k) = f(k) + g(k) = e^k + 3 k + 4\) is neither exponential nor linear.
Differences
If \(f\) and \(g\) are functions, then \(f - g\) is called the difference of \(f\) and \(g\).
The difference of two functions is defined on the intersection of their two domains.
\(f(x) = e^x\) is an exponential function.
\(g(t) = 3 t + 4\) is a linear function.
However, \((f - g)(k) = f(k) - g(k) = e^k - 3 k - 4\) is neither exponential nor linear.
Products
If \(f\) and \(g\) are functions, then \(f \cdot g\), or just \(f g\), is called the product of \(f\) and \(g\).
The product of two functions is defined on the intersection of their two domains.
\(f(x) = e^x\) is an exponential function.
\(g(t) = 3 t + 4\) is a linear function.
However, \((f \cdot g)(k) = f(k) \cdot g(k) = e^k (3 k + 4)\) is neither exponential nor linear.
Quotients
It has been very common to see numbers involved in division: \(15 \div 7\). However, functions are never written as division. They are always written as quotients (fractions).
If \(f\) and \(g\) are functions, then \(\frac {f}{g}\) is called the quotient of \(f\) and \(g\).
The quotient of two functions is defined on the intersection of their two domains
except at the zeros of the function in the denominator.
\(f(x) = e^x\) is an exponential function.
\(g(t) = 3 t + 4\) is a linear function.
However, \(\left (\frac {f}{g}\right )(k) = \frac {f(k)}{g(k)} = \frac {e^k}{(3 k + 4)}\) is neither exponential nor linear.
- Just because you see an exponential formula does not mean you have an
exponential function.
- Just because you see a logarithmic formula does not mean you have a
logarithmic function.
- Just because you see a quadratic formula does not mean you have a
quadratic function.
- Just because you see a linear formula does not mean you have a linear
function.
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more examples can be found by following this link
More Examples of the Function Forms