In VCE Mathematical Methods, we often encounter functions that are combinations of simpler functions. Understanding how to sketch and analyze the graphs of these combined functions is crucial. This section focuses on graphs resulting from the sum, difference, product, and composition of functions, excluding those that result in reciprocal or quotient functions.
The sum of two functions, $f(x)$ and $g(x)$, is defined as:
$$(f + g)(x) = f(x) + g(x)$$
The domain of the sum of two functions is the intersection of their individual domains:
$$dom(f + g) = dom(f) \cap dom(g)$$
The addition of ordinates is a graphical technique used to sketch the sum of two functions. To do this:
If $f(x) = x$ and $g(x) = sin(x)$, then $(f + g)(x) = x + sin(x)$. To sketch this, add the y-values of the line $y = x$ and the sine curve $y = sin(x)$ at various points.
The difference of two functions, $f(x)$ and $g(x)$, is defined as:
$$(f - g)(x) = f(x) - g(x)$$
Similar to the sum, the domain of the difference of two functions is the intersection of their domains:
$$dom(f - g) = dom(f) \cap dom(g)$$
The process is similar to adding ordinates, but you subtract the y-values of $g(x)$ from $f(x)$.
If $f(x) = x^2$ and $g(x) = x$, then $(f - g)(x) = x^2 - x$. This is a parabola. Subtract the y-value of the line $y=x$ from the y-value of the parabola $y=x^2$.
The product of two functions, $f(x)$ and $g(x)$, is defined as:
$$(f \cdot g)(x) = f(x) \cdot g(x)$$
Again, the domain is the intersection of the individual domains:
$$dom(f \cdot g) = dom(f) \cap dom(g)$$
If $f(x) = x$ and $g(x) = e^{-x}$, then $(f \cdot g)(x) = xe^{-x}$. Analyze the sign of $x$ and $e^{-x}$ separately. $e^{-x}$ is always positive, so the sign of the product is determined by the sign of $x$.
The composition of two functions, $f(x)$ and $g(x)$, is defined as:
$$(f \circ g)(x) = f(g(x))$$
This means you substitute the function $g(x)$ into the function $f(x)$.
The domain of the composite function $f(g(x))$ is the set of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.
$$dom(f \circ g) = {x \in dom(g) : g(x) \in dom(f)}$$
Understanding the transformations caused by the inner function $g(x)$ on the outer function $f(x)$ is key. It’s often helpful to consider a table of values.
If $f(x) = \sqrt{x}$ and $g(x) = x - 2$, then $(f \circ g)(x) = \sqrt{x - 2}$. The domain of $g(x)$ is all real numbers. However, the domain of $f(x)$ is $x \geq 0$. Therefore, we must have $x - 2 \geq 0$, which means $x \geq 2$. The domain of $(f \circ g)(x)$ is $[2, \infty)$.
| Operation | Definition | Domain | Graphing Technique |
|---|---|---|---|
| Sum | $(f + g)(x) = f(x) + g(x)$ | $dom(f) \cap dom(g)$ | Addition of Ordinates |
| Difference | $(f - g)(x) = f(x) - g(x)$ | $dom(f) \cap dom(g)$ | Subtraction of Ordinates |
| Product | $(f \cdot g)(x) = f(x) \cdot g(x)$ | $dom(f) \cap dom(g)$ | Analyze signs and key features |
| Composition | $(f \circ g)(x) = f(g(x))$ | ${x \in dom(g) : g(x) \in dom(f)}$ | Understand transformations |
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