# 004 Sample Final A, Problem 1

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Find ${\displaystyle f^{-1}(x)}$ for ${\displaystyle f(x)={\frac {3x-1}{4x+2}}}$

Foundations
How would you find the inverse for a simpler function like ${\displaystyle f(x)=2x+4?}$
You would replace ${\displaystyle f(x)}$ with ${\displaystyle y}$. Then, switch ${\displaystyle x}$ and ${\displaystyle y}$. Finally, we would solve for ${\displaystyle y}$.

Solution:

Step 1:
We start by replacing ${\displaystyle f(x)}$ with ${\displaystyle y}$.
This leaves us with ${\displaystyle y={\frac {3x-1}{4x+2}}}$
Step 2:
Now, we swap ${\displaystyle x}$ and ${\displaystyle y}$ to get ${\displaystyle x={\frac {3y-1}{4y+2}}}$.
Step 3:
Starting with ${\displaystyle x={\frac {3y-1}{4y+2}}}$, we multiply both sides by ${\displaystyle 4y+2}$ to get
${\displaystyle x(4y+2)=3y-1}$.
Now, we need to get all the ${\displaystyle y}$ terms on one side. So, adding ${\displaystyle 1}$ and ${\displaystyle -4xy}$ to both sides we get
${\displaystyle 2x+1=3y-4xy}$.
Step 4:
Factoring out ${\displaystyle y}$, we get ${\displaystyle 2x+1=y(3-4x)}$. Now, dividing by ${\displaystyle (3-4x)}$, we get
${\displaystyle {\frac {2x+1}{3-4x}}=y}$. Replacing ${\displaystyle y}$ with ${\displaystyle f^{-1}(x)}$, we arrive at the final answer
${\displaystyle f^{-1}(x)={\frac {2x+1}{3-4x}}}$
${\displaystyle f^{-1}(x)={\frac {2x+1}{3-4x}}}$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 29, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://wiki.math.ucr.edu/index.php?title=004_Sample_Final_A,_Problem_1&oldid=807", "fetch_time": 1642961443000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2022-05/segments/1642320304309.5/warc/CC-MAIN-20220123172206-20220123202206-00683.warc.gz", "warc_record_offset": 672926184, "warc_record_length": 7501, "token_count": 510, "char_count": 1425, "score": 4.8125, "int_score": 5, "crawl": "CC-MAIN-2022-05", "snapshot_type": "latest", "language": "en", "language_score": 0.7656295895576477, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00043-of-00064.parquet"}