Question

4) Verify or prove that the two functions are inverses of each other.a) f(x)= 4x–2 and x+2/24b) f(x)= -3–9 g(x)= -13x–35) Evaluate each composition.a) f(x)= 4x+3 and g(x)=x^2Find f(g(1)) and g(f(1))b) f(x)=x–1 and g(x)=x^2+2x–8Find f(g(2)) and g(f(2))6) Suppose you drop a stone into a pond. It makes ripples with a radius of r=4t, where r is the radius in inches, and t is time in seconds. a) Express the area of ripple as a function of its radius.b) Express the area of a ripple as a function of time.(I also have an image of my other questions).

Answer 1

Given:

a)

$\begin{gathered} f(x)=4x-2 \\ g(x)=\frac{x+2}{24} \end{gathered}$

To verify two functions are inverses of each other,

Step 1: Plug g(x) into f(x),

$\begin{gathered} \text{ To prove : }f\lbrack g(x)\rbrack=x \\ f\lbrack g(x)\rbrack=f\lbrack\frac{x+2}{24}\rbrack \\ =4(\frac{x+2}{24})-2 \\ =\frac{x+2}{6}-2 \\ =\frac{x+2-12}{6} \\ =\frac{x-10}{6}\ne x \\ It\text{ implies that f(x) and g(x) are not inverses of each other.} \end{gathered}$

b)

$\begin{gathered} f(x)=-3x-9 \\ g(x)=-13x-3 \end{gathered}$

Step 1: plug g(x) into f(x)

$\begin{gathered} To\text{ prove : }f\lbrack g(x)\rbrack=x \\ f\lbrack g(x)\rbrack=f(-13x-3) \\ =-3(-13x-3)-9 \\ =39x+9-9 \\ =39x \\ \ne x \\ It\text{ implies that f(x) and g(x) are not inverses of each other.} \end{gathered}$