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# Functions

Functions map an input to an output. For example, the function f(x) = 2x takes an input, x, and multiplies it by two. An input of x = 2 gives you an output of 4. Learn all about functions.

# Function Misconceptions

$$f$$ and $$g$$ are two functions such that $$(f \circ g)(3)=5$$

What is the value of $$(g \circ f)(3)$$ ?

If $$f$$ is a function with domain $$\mathbb{R}$$ and range $$\mathbb{R}$$, does there exist a function $$g$$ such that $$g(x) = f^{-1}(x)$$ ?

In the above image, the red graph is a graph of $$y=f(x)$$, for some function $$f$$.

What transformation of $$f$$ will produce the blue graph in the above graph?

$$f(x)$$ is a function with domain $$\mathbb{R}.$$

$$g(x)$$ is a horizontal transformation of $$f(x)$$ such that $$g(x) = f(x+k).$$

$$h(x)$$ is a vertical transformation of $$f(x)$$ such that $$h(x) = f(x)+k.$$

If you choose $$f(x)$$ carefully, is it possible that $$g(x)$$ might equal $$h(x)$$ for all $$x$$?

$$f$$ is a function whose domain is $$\mathbb{R}$$.

$$g$$ is a function such that $$g(x)=f^{-1}(x)$$

True or false: The domain of $$g$$ must also be $$\mathbb{R}$$.

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