Evaluate
\[ (T \circ p)\left ( \frac {1}{2} \right ) = \answer {-3} \]
step by step
The elementary functions are too nice.
They don’t do anything interesting.
However, the world of functions is crazy. Functions behave in very unexpected ways, which we’ll never see if we stick to the elementary functions.
For a meaningful investigation of functions, we need to explore other types of functions.
We can create new functions from the elementary functions by creating sums, differences, products, and quotients.
We can create new functions from the elementary functions by creating piecewise defined functions.
We can create new functions from the elementary functions by composing them.
Composition
Let \(f\) be a function with domain \(D_f\).
Let \(g\) be a function with domain \(D_g\).
Then the composition of f and g, \(f \circ g\), is defined as
\[ (f \circ g)(x) = f(g(x)) \]
The value of \(g\) becomes a domain number for \(f\).
The composition is defined on a subset of the domain of \(g\). The composition is defined at those numbers in the domain of \(g\) where the value of \(g\) is in the domain of \(f\).
Let \(f(x) = 3 - x\) with its natural domain.
Let \(g(y) = \ln (y)\) with its natural domain.
Defines \(H = f \circ g\) with its natural domain.
\(H(t) = (f \circ g)(t) = f(g(t))\)
The domain of \(H = f \circ g\) is a subset of the domain of \(g\), which is \((0, \infty )\).
We can only use those numbers where \(ln(t)\) is in the domain of \(f\). However, the domain of \(f\) is all real numbers. So, we can use everything from the domain of \(g\).
The induced domain of \(H = f \circ g\) is the domain of \(g\), which is \((0, \infty )\).
\(K(r) = (g \circ f)(r) = g(f(r))\)
The domain of \(K = g \circ f\) is a subset of the domain of \(f\), which is \((-\infty , \infty )\).
We can only use those numbers where \(f(r)\) is in the domain of \(g\). The domain of \(g\) is \((0, \infty )\).
Therefore, we need \(f(x) = 3 - x > 0\). Or, \(x < 3\).
The induced domain of \(K = g \circ f\) is \((-\infty , 3)\).
These are the numbers in the domain of \(f\) where the value of \(f\) is inside the domain of \(g\).
An induced domain is a domain that is force to happen due to given restrictions.
Let \(T(x) = (3-x)(x-5)\) with its natural domain.
Let \(p(t) = \frac {1}{t}\) with its natural domain.
Defines \(Y = T \circ p\) with its natural domain.
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More Examples of Function Forms