丁香园AV

If \(f\left( x\right) =\dfrac{1}{x-1}\text{,}\) then which of the following is equal to \(f\left( \dfrac{1}{x}\right) ?\)

1. \(f(x)\)

2. \(-f(x)\)

3. \(xf(x)\)

4. \(-xf(x)\)

5. \(\dfrac{f(x)}{x}\)

6. \(-\dfrac{f(x)}{x}\)

If \(f(x)=\dfrac{x}{x+3}\text{,}\) then find and simplify \(\dfrac{f(x)-f(2)}{x-2}\text{.}\)

If \(f(x)=x^2\text{,}\) then find and simplify \(\dfrac{f(3+h)-f(3)}{h}\text{.}\)

What is the domain of

1. \(f(x)=\dfrac{\sqrt{x-2}}{x^2-9}\text{?}\)

2. \(g(x)=\dfrac{\sqrt[3]{x-2}}{x^2-9}\text{?}\)

Suppose that \(f(x)=x^3\) and \(g(x)=x\text{.}\) What is the domain of \(\dfrac{f}{g}\text{?}\)

Suppose that \(f(x)=3x-4\text{.}\) Find a function \(g\) such that \((g\circ f)(x)=5x+2\text{.}\)

Which of the following functions is one-to-one?

1. \(f(x)=x^2+4x+3\)

2. \(g(x)=\vert x\vert+2\)

3. \(h(x)=\sqrt[3]{x+1}\)

4. \(F(x)=\cos x\text{,}\) \(-\pi\leq x\leq\pi\)

5. \(G(x)=e^x+e^{-x}\)

What is the inverse of \(f(x)=\ln\left(\dfrac{e^x}{e^x-1}\right)\text{?}\) What is the domain of \(f^{-1}\text{?}\)

Solve the following equations.

1. \(e^{2-x}=3\)

2. \(e^{x^2}=e^{4x-3}\)

3. \(\ln\left(1+\sqrt{x}\right)=2\)

4. \(\ln(x^2-3)=\ln 2+\ln x\)

Find the exact value of \(\sin^{-1}\left(-\sqrt{2}/2\right)-\cos^{-1}\left(-\sqrt{2}/2\right)\text{.}\)

Find \(\sin^{-1}\left(\sin(23\pi/5)\right)\text{.}\)

It can be proved that \(f(x)=x^3+x+e^{x-1}\) is one-to-one. What is the value of \(f^{-1}(3)\text{?}\)

Sketch the graph of \(f(x)=\left\{ \begin{array}{cc} -x \amp \text{ if } x\leq 0 \\ \tan ^{-1}x \amp \text{ if } x>0 \end{array} \right.\)