To find the compositions of the functions \( f \) and \( g \), we will handle each part separately.
- Function Definitions:
- \( f(x) = \frac{8x - 9}{4} \)
- \( g(x) = \frac{4x + 9}{8} \)
Part 1: Find \( (f \circ g)(x) \)
To find \( (f \circ g)(x) \), we need to evaluate \( f(g(x)) \).
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Substitute \( g(x) \) into \( f(x) \): \[ g(x) = \frac{4x + 9}{8} \] So, we have: \[ f(g(x)) = f\left(\frac{4x + 9}{8}\right) \]
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Now plug \( \frac{4x + 9}{8} \) into the formula for \( f(x) \): \[ f\left(\frac{4x + 9}{8}\right) = \frac{8\left(\frac{4x + 9}{8}\right) - 9}{4} \]
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Simplifying this, we first compute \( 8\left(\frac{4x + 9}{8}\right) \): \[ 8\left(\frac{4x + 9}{8}\right) = 4x + 9 \]
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Then, substituting back in, we have: \[ f\left(\frac{4x + 9}{8}\right) = \frac{4x + 9 - 9}{4} = \frac{4x}{4} = x \]
So, we find: \[ (f \circ g)(x) = x \]
Part 2: Find \( (g \circ f)(x) \)
To find \( (g \circ f)(x) \), we need to evaluate \( g(f(x)) \).
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Substitute \( f(x) \) into \( g(x) \): \[ f(x) = \frac{8x - 9}{4} \] So, we have: \[ g(f(x)) = g\left(\frac{8x - 9}{4}\right) \]
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Plug \( \frac{8x - 9}{4} \) into the formula for \( g(x) \): \[ g\left(\frac{8x - 9}{4}\right) = \frac{4\left(\frac{8x - 9}{4}\right) + 9}{8} \]
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Simplifying this, we first compute \( 4\left(\frac{8x - 9}{4}\right) \): \[ 4\left(\frac{8x - 9}{4}\right) = 8x - 9 \]
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Then, substituting back in, we have: \[ g\left(\frac{8x - 9}{4}\right) = \frac{8x - 9 + 9}{8} = \frac{8x}{8} = x \]
So, we find: \[ (g \circ f)(x) = x \]
Final Answers:
- Part 1: \( (f \circ g)(x) = x \)
- Part 2: \( (g \circ f)(x) = x \)
1 answer