2.2 Functions & Inverses

The function `f`is defined by `f(x)=(4x+1)/(x+4)`, where `x ∈R, x≠-4.`

(a) For the graph of  `f`

      (i) Write down the equation of the vertical asymptote;

      (ii) Find the equation of the horizontal asymptote.

(b) (i) Find `f^(-1) (x).`

      (ii) Using an algebraic approach, show that the graph of `f^(-1)` is obtained by a reflection of the graph of `f` in the `y`-axis followed by a reflection in the `x`-axis.

The graphs of `f`and `f^(-1)` intersect at `x=p` and `x=q`, where `p.

(c) (i) Find the value of `p`and the value of `q`.

      (ii) Hence, find the area enclosed by the graph of `f`and the graph of `f^(-1)`

(a) (i) `x=-4`

     (ii) attempt to substitute into `y=a/c` OR table with large values of `x` OR sketch of `f` showing asymptotic behaviour

          `y=4`

(b) (i) `y=(4x+1)/(x+4)`

     attempt to interchange `x` and `y` (seen anywhere)

     `xy+4y=4x+1` OR `xy+4x=4y+1`

     `xy-4x=1-4y` OR `xy-4y=1-4x`

     `f^(-1) (x)=(1-4x)/(x-4)` (accept `y=(1-4x)/(x-4)`)

   (ii) reflection in `y`-axis given by `f(-x)`

         `f(-x)=(-4x+1)/(-x+4)`

         reflection of their `f(-x)` in `x`-axis given by `-f(-x)` accept “now `-f(x)`"

         `(-f(-x))=(-(-4x+1))/(-x+4)`

         `=(-4x+1)/(x-4)` OR `(4x-1)/(-x+4)`

         `=(1-4x)/(x-4)``(=f^(-1) (x))`

(c) (i) attempt to solve `f(x)=f^(-1) (x)` using graph or algebraically `p=-1` AND `q=1`

     (ii) attempt to set up an integral to find area between `f` and `f^(-1)`

           `∫_(-1)^1((4x+1)/(x+4)-(1-4x)/(x-4)) dx`

           `=0.675231…`

           `=0.675`      

The functions `f` and `g` are defined by `f(x)=2x-x^3` and `g(x)=tanx`.

(a) Find `(f∘g)(x)`

(b) On the following grid, sketch the graph of `y=(f∘g)(x)` for `-1≤x≤1`. 

      Write down and clearly label the coordinates of any local maximum or minimum points.

(a) attempt to substitute `g` into `f`

      `(f∘g)(x)=2tanx-tan^3 x`

(b) 

Consider the function `h(x)=√(4x-2)`, for `x≥1/2`.

(a) For the graph of  

     (i) Find `h^(-1) (x)`, the inverse of `h(x)`, and state its domain.

     (ii) Write down the range of `h^(-1) (x)`.

(b) The graph of `h` intersects the graph of  `h^(-1)` at two points. Find the `x`-coordinates of these two points.

(c) Find the area enclosed by the graph of `h` and the graph of `h^(-1)`.

(d) Find `h^' (x)`.

(e) Find the value of `x` for which the graph of  `h` and the graph of  `h^(-1)` have the same gradient.

(a) (i) swapping `x` and `y`, or `h(h^(-1) (x))=x`

          `h^(-1) (x)=(x^2+2)/4`

          recognizing range of  `h` is domain of  `h^(-1)`

          Domain: `x≥0`

     (ii) range of `h^(-1)` is `y≥1/2`

(b) `√(4x-2)=(x^2+2)/4` OR `√(4x-2)=x` OR `(x^2+2)/4=x`

      `x=0.585786…, x=3.414213… (=2+√2)`

      `x=0.586, x=3.41`

(c) attempt to form integral of the difference between `h(x)` and their `h^(-1)`, using their limits from part (b)

      `∫_(0.585786…)^(3.414213…)(h(x)-h^(-1) (x)) ,dx "OR" ∫_(0.585786…)^(3.414213…)(√(4x-2)-(x^2+2)/4) dx`

      `6.5996632…-4.7140452…`

      `1.88561…`

      area `=1.89`

(d) attempt to use chain rule or power rule

     `h^' (x)=4⋅1/2(4x-2)^(-1⁄2)`

     `h^' (x)=2/√(4x-2)`

(e) `(h^(-1) )^' (x)=x/2`

     equating their `h^' (x)` to the derivative of their `h^(-1) (x)` and attempting to solve for `x`

     `2/√(4x-2)=x/2`

     `1.772776…`

     `x=1.77`

Let `f(x)=ln(x+5)+ln2`, for `x>-5`.

(a) Find `f^(-1) (x)`.

Let `g(x)=e^x`

(b) Find `(g∘f)(x)`, giving your answer in the form `ax+b`, where `a,b∈Z`.

(a) `ln(x+5)+ln2=ln(2(x+5)) (=ln(2x+10))`

     interchanging `x` and `y` (seen anywhere) e.g.`x=ln(2y+10)`

     evidence of correct manipulation e.g. `e^x=2y+10`

     `f^(-1) (x)=(e^x-10)/2`

(b) evidence of composition in correct order e.g. `(g∘f)(x)=g(ln(x+5)+ln2)`

     `=e^(ln(2(x+5)))=2(x+5)`

     `(g∘f)(x)=2x+10`

Let `f(x)=2x^3+3` and `g(x)=e^3x-2`

(a) (i) Find `g(0)`.

     (ii) Find `(f∘g)(0)`

(b) Find `f^(-1) (x)`

(a) (i)  `g(0)=e^0-2`

                  `=-1`

     (ii) attempt to find `(f∘g)(x)`

           e.g. `(f∘g)(x)=f(e^3x-2)=2(e^3x-2)^3+3`

           correct expression for `(f∘g)(x)`

           e.g. `2(e^3x-2)^3+3`

          `(f∘g)(0)=1`

(b) interchanging `x` and `y` (seen anywhere)

     e.g. `x=2y^3+3` 

     attempt to solve

     e.g. `y^3=(x-3)/2`

     `f^(-1) (x)=∛((x-3)/2)`

Let `f(x)=7-2x` and `g(x)=x+3`.

(a) Find `(g∘f)(x)`

(b) Write down `g^(-1) (x)`.

(c) Find `(f∘g^(-1))(5)`

(a) attempt to form composite

     e.g. `g(7-2x)`, `7-2x+3`

    `(g∘f)(x)=10-2x`

(b) `g^(-1) (x)=x-3`

(c) valid approach 

     e.g. `g^(-1) (5)`, `2`, `f(5)`

    `f(2)=3`

 

Let `f(x)=2x-1` and `g(x)=3x^2+2`.

(a) Find `f^(-1) (x)`.

(b) Find `(f∘g)(1)`.

(a) interchanging `x` and `y` (seen anywhere)

     e.g. `x=2y-1`

     correct manipulation

     e.g. `x+1=2y`

    `f^(-1) (x)=(x+1)/2`

(b) attempt to form composite (in any order)

      e.g. `2(3x^2+2)-1`, `3(2x-1)^2+2`

     `(f∘g)(1)=9`

 

Let `f(x)=4/(x+2), x≠-2` and `g(x)=x-1`

If `h=g∘f`, find

(a) `h(x)`

(b) `h^(-1) (x)`, where `h^(-1)` is the inverse of `h`

(a) `h(x)=g(4/(x+2))`

            `=4/(x+2)-1 (=(2-x)/(2+x))`

(b) `x=4/(y+2)-1 "(interchanging " x" and " y)`

      Attempting to solve for `y`:

     `(y+2)(x+1)=4 (⇒y+2=4/(x+1))`

     `h^(-1) (x)=4/(x+1)-2 (x≠-1)`

Consider the functions given below.

                                                       `f(x)=2x+3`

                                                       `g(x)=1/x, x≠0`

(a) (i) Find  `(g∘f)(x)` and write down the domain of the function.

     (ii) Find `(f∘g)(x)` and write down the domain of the function.

(b) Find the coordinates of the point where the graph of `y=f(x)` and the graph of `y=(g^(-1)∘f∘g)(x)`intersect.

(a) (i) `(g∘f)(x)=1/(2x+3), x≠-3/2 " (or equivalent)"`

     (ii) `(f∘g)(x)=2/x+3, x≠0" (or equivalent)"`

(b) `f(x)=(g^(-1)∘f∘g)(x)⇒(g∘f)(x)=(f∘g)(x)`

     `1/(2x+3)=2/x+3`

     `6x^2+12x+6=0" (or equivalent)"`

     `x=-1, y=1 "(coordinates are "(-1,1)")"`

Consider the following functions:

                                                   `f(x)=(2x^2+3)/75, x≥0`

                                                   `g(x)=(|3x-4|)/10, x∈R.`

(a)  State the range of `f` and of `g`

(b) Find an expression for the composite function `f∘g(x)` in the form `(ax^2+bx+c)/3750,` where `a,b` and `c∈Z`

(c) (i) Find an expression for the inverse function `f^(-1) (x)`

     (ii) State the domain and range of `f^(-1)`

The domains of `f` and `g` are now restricted to `{0,1,2,3,4}.`

(d) By considering the values of `f` and `g` on this new domain, determine which of `f` and `g`could be used to find a probability distribution for a discrete random variable `X`, stating your reasons clearly.

(e) Using this probability distribution, calculate the mean of `X`

(a) `f (x)≥1/25`

     `g(x)∈R, g(x)≥0`

(b) `f∘g(x)=(2((3x-4)/10)^2+3)/75`

     `=(2(9x^2-24x+16))/100+3/75`

     `=(9x^2-24x+166)/3750`

(c) (i) `y=(2x^2+3)/75`

          `x^2=(75y-3)/2`

          `x=√((75y-3)/2)`

          `⇒f^(-1) (x)=√((75x-3)/2)`

     (ii) domain: `x≥1/25`; range: `f^(-1) (x)≥0`

(d) probabilities from `f(x)`     

     probabilities from `g(x)`

 only in the case of `f(x)` does `∑P(X=x)=1`, hence only `f(x)` can be used as a probability mass function.

(e) `E(x)=∑x⋅P(X=x)`

     `=5/75+22/75+63/75+140/75=230/75=(46/15)`