# Functional equation dependent on a constant

## Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$
i) $f(ax) = a^2f(x)$ and
ii) $f(f(x)) = a f(x).$[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.26.6" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" hover_enabled="0"][et_pb_accordion_item title="Source of the problem" open="off" _builder_version="3.26.6"]Baltic Way 2016[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="4.3.2" hover_enabled="0" open="on"]

### Functional Equation

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.26.6" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.26.6"]Show that the choices $a=0,1$ work.

[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.26.6"]Show that $af(f(x))=a^2f(f(x))$. As we have already dealt with $a=0$, this gives $af(f(x))=f(f(x))$. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.26.6"]Hint 3 gives $(a-1)f(f(x))=0$. As $a=1$ has already been dealt with, we must consider the option $f(f(x))\equiv 0$.[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.26.6"]Hint 3 gives $af(x)\equiv 0$. As $a\neq 0$, we have $f(x)\equiv 0$. This contradicts the fact that $f$ is non-constant. Hence, $a=0,1$ are the only options.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" inline_fonts="Aclonica"]

## Watch video

[/et_pb_text][et_pb_code _builder_version="3.26.4"]

## Similar Problems

[/et_pb_text][et_pb_text _builder_version="4.3.2"]

https://www.cheenta.com/average-amc-10b-2019-problem-no-4/

[/et_pb_text][et_pb_post_slider include_categories="9" _builder_version="3.22.4"][/et_pb_post_slider][et_pb_divider _builder_version="3.22.4" background_color="#0c71c3"][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]

This site uses Akismet to reduce spam. Learn how your comment data is processed.

### Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.