Question
In which choice is \( y \) a nonlinear function of \( x \) ? A \( y=\frac{x}{5}-3 \) B \( y=4-3 x \) C \( y=\frac{6}{x-4}+7 \) D \( y=\frac{x+1}{8}+9 \)
Ask by Deleon Schofield. in the United States
Feb 27,2025
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Tutor-Verified Answer
Answer
Choice C: \( y = \frac{6}{x-4} + 7 \) is a nonlinear function of \( x \).
Solution
To determine which choice has \( y \) as a nonlinear function of \( x \), we need to analyze each equation.
1. **Choice A:** \( y = \frac{x}{5} - 3 \)
- This is a linear function because it can be expressed in the form \( y = mx + b \) where \( m = \frac{1}{5} \) and \( b = -3 \).
2. **Choice B:** \( y = 4 - 3x \)
- This is also a linear function, as it can be rewritten as \( y = -3x + 4 \).
3. **Choice C:** \( y = \frac{6}{x-4} + 7 \)
- This function is nonlinear because it involves a variable in the denominator, which creates a hyperbola.
4. **Choice D:** \( y = \frac{x+1}{8} + 9 \)
- This is a linear function, as it can be expressed as \( y = \frac{1}{8}x + \frac{1}{8} + 9 \).
From this analysis, the only choice where \( y \) is a nonlinear function of \( x \) is:
**Choice C: \( y = \frac{6}{x-4} + 7 \)**.
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Simplify this solution Extra Insights
Let's break it down! The key to spotting a nonlinear function is to look for variables in the denominator or any variables raised to a power other than 1. The third choice, \( y = \frac{6}{x-4} + 7 \), is the standout. The presence of \( x \) in the denominator makes it a nonlinear function. The other choices represent either linear functions or constant transformations of linear functions. So, to sum it up succinctly: if you see a variable in a denominator or any higher power involved, you’ve found a nonlinear gem!
