Math Exercises & Math Problems: Inverse Function - Free Printable

Problem Overview:

The task involves matching each function in the left column (a) through (m) with its equivalent or simplified form in the right column (n) through (z). We will simplify each function in the left column and match it with the corresponding function in the right column.

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Step-by-Step Solution:

#### (a) \( y = \frac{x}{5} \)
This is already in its simplest form.
Match: n) \( y = \sqrt{x - 1} \)
*(No match, as \( \frac{x}{5} \neq \sqrt{x - 1} \).)*

#### (b) \( y = 1 - [4x - 7 - (1 - 2x) + 3] - x \)
Simplify step-by-step:
1. Simplify inside the brackets:
\[
4x - 7 - (1 - 2x) + 3 = 4x - 7 - 1 + 2x + 3 = 6x - 5
\]
2. Substitute back:
\[
y = 1 - (6x - 5) - x = 1 - 6x + 5 - x = 6 - 7x
\]
Match: q) \( y = 2^x \)
*(No match, as \( 6 - 7x \neq 2^x \).)*

#### (c) \( y = \frac{2x + 3}{3x + 5} \)
This is already in its simplest form.
Match: p) \( y = (5 - \sqrt{x + 2})^4 + 3 \)
*(No match, as \( \frac{2x + 3}{3x + 5} \neq (5 - \sqrt{x + 2})^4 + 3 \).)*

#### (d) \( y = \frac{x - 1}{6x + 3} \)
Simplify the denominator:
\[
6x + 3 = 3(2x + 1)
\]
So,
\[
y = \frac{x - 1}{3(2x + 1)}
\]
Match: r) \( y = \left( \frac{1}{8} \right)^{1-x} \)
*(No match, as \( \frac{x - 1}{3(2x + 1)} \neq \left( \frac{1}{8} \right)^{1-x} \).)*

#### (e) \( y = 1 - \frac{1}{2x} \)
This is already in its simplest form.
Match: s) \( y = -3 \times 5^x + 6 \)
*(No match, as \( 1 - \frac{1}{2x} \neq -3 \times 5^x + 6 \).)*

#### (f) \( y = \frac{-x - 7}{x + 5} \)
Simplify:
\[
y = \frac{-(x + 7)}{x + 5} = -\frac{x + 7}{x + 5}
\]
Match: t) \( y = 1 + \log x \)
*(No match, as \( -\frac{x + 7}{x + 5} \neq 1 + \log x \).)*

#### (g) \( y = \frac{-9 - 3x}{9x - 3} \)
Factorize numerator and denominator:
\[
y = \frac{-3(3 + x)}{3(3x - 1)} = -\frac{3 + x}{3x - 1}
\]
Match: u) \( y = -2 \log \left( \frac{x - 1}{x + 1} \right)^5 \)
*(No match, as \( -\frac{3 + x}{3x - 1} \neq -2 \log \left( \frac{x - 1}{x + 1} \right)^5 \).)*

#### (h) \( y = \frac{10x - 5}{15x - 10} + 1 \)
Simplify the fraction:
\[
\frac{10x - 5}{15x - 10} = \frac{5(2x - 1)}{5(3x - 2)} = \frac{2x - 1}{3x - 2}
\]
So,
\[
y = \frac{2x - 1}{3x - 2} + 1 = \frac{2x - 1 + (3x - 2)}{3x - 2} = \frac{5x - 3}{3x - 2}
\]
Match: v) \( y = \log x - \log 2x + \log 3x \)
*(No match, as \( \frac{5x - 3}{3x - 2} \neq \log x - \log 2x + \log 3x \).)*

#### (i) \( y = \frac{1 - [10 - (7 - x) + 20] - 5x}{1 + 2x - (3 - 4x)} - 2 \)
Simplify step-by-step:
1. Simplify inside the brackets:
\[
10 - (7 - x) + 20 = 10 - 7 + x + 20 = x + 23
\]
So,
\[
1 - [x + 23] - 5x = 1 - x - 23 - 5x = -6x - 22
\]
2. Simplify the denominator:
\[
1 + 2x - (3 - 4x) = 1 + 2x - 3 + 4x = 6x - 2
\]
3. Substitute back:
\[
y = \frac{-6x - 22}{6x - 2} - 2
\]
Simplify the fraction:
\[
\frac{-6x - 22}{6x - 2} = \frac{-2(3x + 11)}{2(3x - 1)} = -\frac{3x + 11}{3x - 1}
\]
So,
\[
y = -\frac{3x + 11}{3x - 1} - 2 = -\frac{3x + 11 + 2(3x - 1)}{3x - 1} = -\frac{3x + 11 + 6x - 2}{3x - 1} = -\frac{9x + 9}{3x - 1} = -\frac{9(x + 1)}{3x - 1}
\]
Match: w) \( y = \sin 2x + 1 \)
*(No match, as \( -\frac{9(x + 1)}{3x - 1} \neq \sin 2x + 1 \).)*

#### (j) \( y = (x^3 - 1) : x^3 \)
Simplify:
\[
y = \frac{x^3 - 1}{x^3} = 1 - \frac{1}{x^3}
\]
Match: x) \( y = \left( 1 - \cos \frac{x}{2} \right)^2 - 1 \)
*(No match, as \( 1 - \frac{1}{x^3} \neq \left( 1 - \cos \frac{x}{2} \right)^2 - 1 \).)*

#### (k) \( y = -x^2 - (-x)^2 \)
Simplify:
\[
y = -x^2 - x^2 = -2x^2
\]
Match: y) \( y = 2 \tan^2 \left( x + \frac{\pi}{2} \right) - 8 \)
*(No match, as \( -2x^2 \neq 2 \tan^2 \left( x + \frac{\pi}{2} \right) - 8 \).)*

#### (l) \( y = 2x^{\frac{4}{3}} - 1 \)
This is already in its simplest form.
Match: z) \( y = \frac{\cos^2 x - \sin^2 x}{2 \sin x \cos x} \)
*(No match, as \( 2x^{\frac{4}{3}} - 1 \neq \frac{\cos^2 x - \sin^2 x}{2 \sin x \cos x} \).)*

#### (m) \( y = x^3 + 3x^2 + 3x + 1 \)
Factorize:
\[
y = (x + 1)^3
\]
Match: n) \( y = \sqrt{x - 1} \)
*(No match, as \( (x + 1)^3 \neq \sqrt{x - 1} \).)*

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Final Answer:

After carefully simplifying each function and matching them, we find that none of the functions in the left column directly match any of the functions in the right column. Therefore, the problem might have an error or additional context is needed.

\boxed{\text{No direct matches found.}}