Chapter 6, Problem 7CA

To determine

To select: all those functions whose $x-$ value is an integer when $f(x)=10$ from the given functions.

Answer to Problem 7CA

The required functions that satisfies given condition are $f(x)=3x-2,f(x)=-2x+4,f(x)=\frac{3}{2}x+4,f(x)=\frac{1}{2}x-6,f(x)=4x+14$ of $f\left(-7\right)=e^7=1096.63$

Explanation of Solution

Given information :

Given list of functions

  

Given condition is $f(x)=10$ for an integer $x$

  $\begin{array}{l}i.Letf(x)=10\\ \Rightarrow 10=3x-2\\ \Rightarrow 3x=10+2\\ \Rightarrow 3x=12\\ \therefore x=2\text{is}\text{an}\text{integer}\text{.}\end{array}$

Thus, $f(x)=3x-2$ satisfies given condition.

Similarly way for other functions,

  $\begin{array}{l}\text{consider}-2x+4=10\\ \Rightarrow 2x=4-10\\ \Rightarrow 2x=-6\\ \therefore x=-3\text{is}\text{an}\text{integer}.\end{array}$

Thus, $f(x)=-2x+4$ satisfies given condition.

  $\begin{array}{l}\text{consider}\frac{3}{2}x+4=10\\ \Rightarrow \frac{3}{2}x=10-4\\ \Rightarrow \frac{3}{2}x=6\\ \therefore x=4\text{is}\text{an}\text{integer}.\end{array}$

Thus, $f(x)=\frac{3}{2}x+4$ satisfies given condition

  $\begin{array}{l}\text{consider}-3x+5=10\\ \Rightarrow 3x=5-10\\ \Rightarrow 3x=-5\\ \therefore x=\frac{-5}{3}\text{is}\text{not}\text{an}\text{integer}\text{.}\end{array}$

Thus, $f(x)=-3x+5$ not satisfies given condition.

  $\begin{array}{l}\text{consider}\frac{1}{2}x-6=10\\ \Rightarrow \frac{1}{2}x=10+6\\ \Rightarrow x=16\times 2\\ \therefore x=32\text{is}\text{an}\text{integer}\text{.}\end{array}$

Thus, $f(x)=\frac{1}{2}x-6$ satisfies given condition.

  $\begin{array}{l}\text{consider}4x+14=10\\ \Rightarrow 4x=10-14\\ \Rightarrow 4x=-4\\ \therefore x=-1\text{is}\text{an}\text{integer}\text{.}\end{array}$

Thus, $4x+14$ satisfies given condition.

Hence, the required functions that satisfies given condition are $f(x)=3x-2,f(x)=-2x+4,f(x)=\frac{3}{2}x+4,f(x)=\frac{1}{2}x-6,f(x)=4x+14$