Symmetry and verification by hand

Medium0 of 18 parts

The first group asks you to substitute $-x$ into a rule and simplify, which is the algebra behind deciding whether a function is even or odd. Give the answer as a rule in $x$ .

The second group asks for a composition as a rule. When two functions are inverses, both compositions collapse to $x$ , and typing x is the whole answer. When they are not, the leftover tells you by how much they miss.

Type sqrt(x) for a square root and ^ for a power. Marking is by agreement, so $(2x+3)^2$ and $4x^2 + 12x + 9$ both count.

For $f(x) = x^4 - 2x^2$ , write $f(-x)$ as a rule in $x$ , simplified.

in terms of x

For $f(x) = x^3 - 4x$ , write $f(-x)$ as a rule in $x$ , simplified.

in terms of x

For $f(x) = x^3 - 4x$ , write $-f(x)$ as a rule in $x$ , simplified.

in terms of x

For $f(x) = x^2 + x$ , write $f(-x)$ as a rule in $x$ , simplified.

in terms of x

For $f(x) = (x - 2)^2$ , write $f(-x)$ as a rule in $x$ .

in terms of x

$f$ is odd and $f(-2) = 5$ . Find $f(2)$ .

$f$ is even and $f(4) = -3$ . Find $f(-4)$ .

$f$ is odd and 0 is in its domain. Find $f(0)$ .

With $f(x) = 4x - 1$ and $g(x) = \dfrac{x + 1}{4}$ , write $f(g(x))$ as a rule in $x$ , simplified.

in terms of x

With the same $f$ and $g$ , write $g(f(x))$ as a rule in $x$ , simplified.

in terms of x

With $f(x) = \dfrac{x}{3} + 2$ and $g(x) = 3x - 6$ , write $f(g(x))$ as a rule in $x$ , simplified.

in terms of x

With $f(x) = 2x + 5$ and $h(x) = \dfrac{x - 5}{2}$ , write $h(f(x))$ as a rule in $x$ , simplified.

in terms of x

With $f(x) = 2x + 5$ and $k(x) = \dfrac{x + 5}{2}$ , write $f(k(x))$ as a rule in $x$ , simplified.

in terms of x

With $f(x) = \sqrt{x}$ and $g(x) = x - 4$ , write $f(g(x))$ as a rule in $x$ .

in terms of x

With $f(x) = x^2$ and $g(x) = x - 5$ , find $(f \circ g)(2)$ .

With the same $f$ and $g$ , find $(g \circ f)(2)$ .

With $f(x) = x + 3$ and $g(x) = 4x$ , write $(f \circ g)(x)$ as a rule in $x$ .

in terms of x

With the same $f$ and $g$ , write $(g \circ f)(x)$ as a rule in $x$ .

in terms of x