Practice evaluating rational functions with this printable worksheet featuring algebraic substitution problems.

Let’s solve each part step by step.

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## A) Evaluate each function at the specified value.

1) \( f(x) = \frac{x + 1}{3} \); \( x = -10 \)

Substitute \( x = -10 \):

\[
f(-10) = \frac{-10 + 1}{3} = \frac{-9}{3} = -3
\]

✔ Answer: -3

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2) \( f(x) = \frac{x^2 - 2x}{4x} \); \( x = 4 \)

Substitute \( x = 4 \):

Numerator: \( 4^2 - 2(4) = 16 - 8 = 8 \)

Denominator: \( 4 \cdot 4 = 16 \)

\[
f(4) = \frac{8}{16} = \frac{1}{2}
\]

✔ Answer: \( \frac{1}{2} \)

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3) \( f(x) = \frac{5x - 8}{-10x + 4} \); \( x = -2 \)

Substitute \( x = -2 \):

Numerator: \( 5(-2) - 8 = -10 - 8 = -18 \)

Denominator: \( -10(-2) + 4 = 20 + 4 = 24 \)

\[
f(-2) = \frac{-18}{24} = -\frac{3}{4} \quad \text{(simplified)}
\]

✔ Answer: \( -\frac{3}{4} \)

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4) \( f(x) = \frac{3x + 1}{5} \); \( x = 8 \)

Substitute \( x = 8 \):

\[
f(8) = \frac{3(8) + 1}{5} = \frac{24 + 1}{5} = \frac{25}{5} = 5
\]

✔ Answer: 5

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## B) Evaluate each function.

1) If \( f(x) = \frac{x^2 - 4x + 6}{2x} \), find \( f(7) \)

Substitute \( x = 7 \):

Numerator: \( 7^2 - 4(7) + 6 = 49 - 28 + 6 = 27 \)

Denominator: \( 2 \cdot 7 = 14 \)

\[
f(7) = \frac{27}{14}
\]

✔ Answer: \( \frac{27}{14} \)

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2) If \( f(x) = \frac{8x}{x - 5} \), find \( f(1) \)

Substitute \( x = 1 \):

\[
f(1) = \frac{8(1)}{1 - 5} = \frac{8}{-4} = -2
\]

✔ Answer: -2

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3) If \( f(x) = \frac{5}{x + 12} \), find \( f(-3) \)

Substitute \( x = -3 \):

\[
f(-3) = \frac{5}{-3 + 12} = \frac{5}{9}
\]

✔ Answer: \( \frac{5}{9} \)

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4) If \( f(x) = \frac{15}{(x + 1)(x - 1)} \), find \( f(0) \)

Substitute \( x = 0 \):

\[
f(0) = \frac{15}{(0 + 1)(0 - 1)} = \frac{15}{(1)(-1)} = \frac{15}{-1} = -15
\]

✔ Answer: -15

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## C) What is the value of \( f(-9) \) if \( f(x) = \frac{7 - x^2}{-8 - x} \)?

We are given:

\[
f(x) = \frac{7 - x^2}{-8 - x}
\]

Plug in \( x = -9 \):

Numerator: \( 7 - (-9)^2 = 7 - 81 = -74 \)

Denominator: \( -8 - (-9) = -8 + 9 = 1 \)

\[
f(-9) = \frac{-74}{1} = -74
\]

✔ Answer: i) -74

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## ✔ Final Answers:

A)
1) -3
2) \( \frac{1}{2} \)
3) \( -\frac{3}{4} \)
4) 5

B)
1) \( \frac{27}{14} \)
2) -2
3) \( \frac{5}{9} \)
4) -15

C)
i) -74

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Let me know if you’d like to see simplifications or domain restrictions too!