Functions

Function Misconceptions

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

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

\frac{1}{5}

-\frac{1}{5}

-5 It is impossible to determine with the information given

by Brilliant Staff

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

Always Not necessarily Never

by Brilliant Staff

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?

y=f(x-3)

y=f(x)+3

y=f(x+3)

y=f(x)-3

by Brilliant Staff

$f(x)$ 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 $x$ ?

There are no functions,

f(x),

for which it's possible. There is one unique,

f(x),

for which it's possible. There are infinitely many functions,

f(x),

for which it's possible.

by Brilliant Staff

$f$ 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 $\mathbb{R}$ .

True False

by Brilliant Staff