Answer :
Sure, let's analyze each given statement for the function [tex]\( f(x) = \frac{10}{x+5} \)[/tex], one by one:
1. The domain of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, -5) \cup (-5, \infty) \)[/tex].
This is true. The function [tex]\( f(x) = \frac{10}{x+5} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex] except where the denominator is zero, which is [tex]\( x = -5 \)[/tex]. Therefore, the domain is all real numbers except [tex]\( x = -5 \)[/tex].
2. The range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 2) \cup (2, \infty) \)[/tex].
This is false. The function [tex]\( f(x) \)[/tex] can take on any real value except zero, but the range given does not accurately describe this behavior. The correct range would not match any of the provided options.
3. The [tex]\( x \)[/tex]-intercept is at [tex]\((-5,0)\)[/tex].
This is false. To find the [tex]\( x \)[/tex]-intercept, we need to solve [tex]\( f(x) = 0 \)[/tex]. For [tex]\( \frac{10}{x+5} = 0 \)[/tex], there is no real solution as the numerator is a constant and cannot be zero. Thus, the function has no [tex]\( x \)[/tex]-intercept.
4. The [tex]\( y \)[/tex]-intercept is [tex]\((0,2)\)[/tex].
This is true. To find the [tex]\( y \)[/tex]-intercept, we substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = \frac{10}{0+5} = \frac{10}{5} = 2. \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\((0, 2)\)[/tex].
5. There is a vertical asymptote at [tex]\( x = -5 \)[/tex].
This is true. A vertical asymptote occurs where the denominator is zero (and thus the function goes to infinity). Since the denominator is zero at [tex]\( x = -5 \)[/tex], there is a vertical asymptote there.
6. The end behavior is [tex]\( x \rightarrow -\infty, f(x) \rightarrow 0 \)[/tex] and [tex]\( x \rightarrow \infty, f(x) \rightarrow 0 \)[/tex].
This is true. As [tex]\( x \)[/tex] approaches either infinity or negative infinity, the value of [tex]\( f(x) \)[/tex] tends towards zero because the numerator is a constant and the denominator increases in magnitude.
By analyzing each statement, we conclude the statuses as follows:
1. Domain: True ✅
2. Range: False ❌
3. [tex]\( x \)[/tex]-intercept: False ❌
4. [tex]\( y \)[/tex]-intercept: True ✅
5. Vertical asymptote: True ✅
6. End behavior: True ✅
1. The domain of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, -5) \cup (-5, \infty) \)[/tex].
This is true. The function [tex]\( f(x) = \frac{10}{x+5} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex] except where the denominator is zero, which is [tex]\( x = -5 \)[/tex]. Therefore, the domain is all real numbers except [tex]\( x = -5 \)[/tex].
2. The range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 2) \cup (2, \infty) \)[/tex].
This is false. The function [tex]\( f(x) \)[/tex] can take on any real value except zero, but the range given does not accurately describe this behavior. The correct range would not match any of the provided options.
3. The [tex]\( x \)[/tex]-intercept is at [tex]\((-5,0)\)[/tex].
This is false. To find the [tex]\( x \)[/tex]-intercept, we need to solve [tex]\( f(x) = 0 \)[/tex]. For [tex]\( \frac{10}{x+5} = 0 \)[/tex], there is no real solution as the numerator is a constant and cannot be zero. Thus, the function has no [tex]\( x \)[/tex]-intercept.
4. The [tex]\( y \)[/tex]-intercept is [tex]\((0,2)\)[/tex].
This is true. To find the [tex]\( y \)[/tex]-intercept, we substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = \frac{10}{0+5} = \frac{10}{5} = 2. \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\((0, 2)\)[/tex].
5. There is a vertical asymptote at [tex]\( x = -5 \)[/tex].
This is true. A vertical asymptote occurs where the denominator is zero (and thus the function goes to infinity). Since the denominator is zero at [tex]\( x = -5 \)[/tex], there is a vertical asymptote there.
6. The end behavior is [tex]\( x \rightarrow -\infty, f(x) \rightarrow 0 \)[/tex] and [tex]\( x \rightarrow \infty, f(x) \rightarrow 0 \)[/tex].
This is true. As [tex]\( x \)[/tex] approaches either infinity or negative infinity, the value of [tex]\( f(x) \)[/tex] tends towards zero because the numerator is a constant and the denominator increases in magnitude.
By analyzing each statement, we conclude the statuses as follows:
1. Domain: True ✅
2. Range: False ❌
3. [tex]\( x \)[/tex]-intercept: False ❌
4. [tex]\( y \)[/tex]-intercept: True ✅
5. Vertical asymptote: True ✅
6. End behavior: True ✅
