The graphs of f and g are shown. Find the function values...

The graphs of $f$ and $g$ are shown. Find the function values for the given values of $x$, if possible.
a. $(f+g)(0)$
b. $(g-f)(2)$
c. $(g \cdot f)(-1)$
d. $\left(\frac{g}{f}\right)(1)$
e. $(f \circ g)(4)$
f. $(g \circ f)(0)$
g. $g(f(4))$

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Key Concepts

Arithmetic Operations on Functions

This concept involves combining two functions using operations such as addition, subtraction, multiplication, and division. To evaluate expressions like (f+g)(x), (g-f)(x), or (g·f)(x), one calculates the value of each function at x individually and then performs the appropriate arithmetic operation on these values.

Function Composition

Function composition is the process of applying one function to the result of another, expressed as (f ? g)(x) or (g ? f)(x). This requires first evaluating the inner function at x and then using that result as the input for the outer function.

Graphical Interpretation

When functions are defined by their graphs, one must read the output value corresponding to a given input by locating the point on the graph where the input is located. This skill enables one to determine exact function values without an explicit algebraic formula.

Domain and Range Considerations

Understanding the domains and ranges of functions is crucial, especially when performing operations or compositions, as certain values might be undefined (for example, division by zero) or fall outside the domain of one of the functions. This concept ensures that evaluations and operations are performed only where the functions are valid.

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