0. Functions

Determine whether the following statements are true and give an explanation or counterexample.

If y= 3ˣ , then x = ³√y

Determine whether the following statements are true and give an explanation or counterexample.

$\frac{{\log }_{b}x}{{\log }_{b}y}={\log }_{b}x-{\log }_{b}y$

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

{Use of Tech} $f\(\left\)(x\(\right\))=\(\frac{1}{x^2}\)$

Find the inverse function (on the given interval, if specified) and graph both $f$ and $f^{-1}$ on the same set of axes. Check your work by looking for the required symmetry in the graphs.

$f\(\left\)(x\(\right\))=8-4x$

Find the inverse function (on the given interval, if specified) and graph both $f$ and $f^{-1}$ on the same set of axes. Check your work by looking for the required symmetry in the graphs.

$f\(\left\)(x\(\right\))=x^2+4$, for $x\(\geq{0}\)$

Find the inverse $f^{-1}\(\left\)(x\(\right\))$ of each function (on the given interval, if specified).

$f\(\left\)(x\(\right\))=\(\frac{2}{x^2+1}\)$, for $x\(\geq{0}\)$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f\left(x\right)=x^2+1$ , then $f^{-1}\left(x\right)=\frac{1}{x^2+1}$.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f\left(x\right)=\frac{1}{x}$ , then $f^{-1}\left(x\right)=\frac{1}{x}$

Find the inverse $f^{-1}\(\left\)(x\(\right\))$ of each function (on the given interval, if specified).

$f\(\left\)(x\(\right\))=10^{-2x}$

Find the inverse $f^{-1}\(\left\)(x\(\right\))$ of each function (on the given interval, if specified).

$f\(\left\)(x\(\right\))=\(\frac{x}{x-2}\)$, for $x\(\gt{2}\)$

Finding inverses Find the inverse function.

ƒ(x) = 3x - 4

Finding inverses Find the inverse function.

ƒ(x) = 3x² + 1, for x ≤ 0

Inverse of composite functions

a. Let g(x) = 2x + 3 and h(x) = x³. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Inverse of composite functions

b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Splitting up curves The unit circle x² + y² = 1  consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x)  (see figure) <IMAGE>.

a. Find the domain and a formula for each function.