Examples of output in the following topics:
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- A function maps a set of inputs onto a set of permissible outputs.
- Each input corresponds with one and only one output
- In the example above, the argument is $x=-3$ and the output is $9$.
- In a function every input number is associated with exactly one output number In a relation an input number may be associated with multiple or no output numbers.
- A function $f$ takes an input $x$ and returns an output $f(x)$.
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- The domain of a function is the set of all possible input values that produce some output value range
- The function provides an output value, $f(x)$, for each member of the domain.
- The rule for a function is that for each input there is exactly one output.
- (Note that although the output value of $1$ repeats, only the input values can not repeat)
- It shows that for every input there is exactly one output value.
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- It is linear - the exponent of the $x$ term is a one (first power), and it follows the definition of a function - for each input ($x$) there is exactly one output ($y$).
- For example, the graph of the equation $x=4$ includes the same input value of $4$ for all points on the line, but would have different output values, such as $(4,-2),(4,0),(4,1),(4,5),$ etcetera.
- Vertical lines are NOT functions, however, since each input is related to more than one output.
- A graph of the equation $y=6$ includes the same output value of 6 for all input values on the line, such as $(-2,6),(0,6),(2,6),(6,6)$, etcetera.
- Horizontal lines ARE functions because the relation (set of points) has the characteristic that each input is related to exactly one output.
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- The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions.
- Less formally, the composition has to make sense in terms of inputs and outputs.
- To evaluate $f(g(3))$, first substitute, or input the value of $3$ into $g(x)$ and find the output.
- Find $f(3)$ and then use that output value as the input value into the $g(x)$ function:
- Use the resulting output as the input to the outside function.
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- Graphs are a great visual representation of functions, showing the relationship between the input and output values as lines or curves
- Since the output, or dependent variable is $y$, for function notation often times $f(x)$ is thought of as $y$.
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- The range of a function is the set of results, solutions, or 'output' values $(y)$ to the equation for a given input.
- If we now look at the possible outputs or $y$-values, $f(x)$, (looking up and down the $y$-axis, notice that the red graph does NOT include $y$-values that are negative, whereas the blue graph does include both positive and negative values.
- The orange graph is the trigonometric function $f(x)=sin (x)$ with a domain of $\mathbb{R}$ and a restricted range of $-1 \leq y \leq 1$ (output values only exist between $-1$ to $1$.
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- With a linear function, each input has an individual, unique output (assuming the output is not a constant).
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- Remember that an inverse function reverses the inputs and outputs.
- Since the function $f(x)=3x^2-1$ has multiple outputs, its inverse is actually NOT a function.
- Let's see what happens when we switch the input and output values and solve for $y$.
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- Each unique input must have a unique output so the function cannot be one-to-one.
- Because each unique input does not have a unique output, this function cannot be one-to-one.
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- In mathematics, a piecewise function
is a function in which more than one formula is used to define the output over different pieces of the domain.
- For all $x$-values less than zero, the first function $(-x)$ is used, which negates the sign of the input value, making the output values positive.
- For all values of $x$ greater than or equal to zero, the second function $(x)$ is used, making the output values equal to the input values.