Certainly! Let's work through each part step-by-step based on the given function [tex]\( f(x) = \frac{4}{x} \)[/tex].
### (a) [tex]\( f(1) \)[/tex]
To find [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = \frac{4}{1} = 4 \][/tex]
### (b) [tex]\( f(2) \)[/tex]
To find [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = \frac{4}{2} = 2 \][/tex]
### (c) [tex]\( f(-1) \)[/tex]
To find [tex]\( f(-1) \)[/tex]:
[tex]\[ f(-1) = \frac{4}{-1} = -4 \][/tex]
### (d) [tex]\( f(\sqrt{2}) \)[/tex]
To find [tex]\( f(\sqrt{2}) \)[/tex]:
[tex]\[ f(\sqrt{2}) = \frac{4}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \][/tex]
### (e) [tex]\( f\left(\frac{1}{x}\right) \)[/tex]
To find [tex]\( f\left(\frac{1}{x}\right) \)[/tex]:
[tex]\[ f\left(\frac{1}{x}\right) = \frac{4}{\frac{1}{x}} = 4 \cdot x = 4x \][/tex]
### (f) [tex]\( f(2x) \)[/tex]
To find [tex]\( f(2x) \)[/tex]:
[tex]\[ f(2x) = \frac{4}{2x} = \frac{4}{2} \cdot \frac{1}{x} = 2 \cdot \frac{1}{x} = \frac{2}{x} \][/tex]
So, the values are:
(a) [tex]\( f(1) = 4 \)[/tex]
(b) [tex]\( f(2) = 2 \)[/tex]
(c) [tex]\( f(-1) = -4 \)[/tex]
(d) [tex]\( f(\sqrt{2}) = 2\sqrt{2} \)[/tex]
(e) [tex]\( f\left(\frac{1}{x}\right) = 4x \)[/tex]
(f) [tex]\( f(2x) = \frac{2}{x} \)[/tex]
