Question

4 select the correct answer. which exponential function does not have an x - intercept? a. f(x)=5^{x - 5}-1 b. f(x)=5^{x - 5}-5 c. f(x)=-5^{x - 5}+5 d. f(x)=-5^{x - 5}-1

Explanation:

Step1: Recall the definition of x - intercept

Set \(y = f(x)=0\) and solve for \(x\). For an exponential function \(a^x\) (\(a>0,a
eq1\)), \(a^x>0\) for all real - valued \(x\).

Step2: Analyze option A

Set \(f(x)=5^{x - 5}-1 = 0\). Then \(5^{x - 5}=1=5^0\). By the one - to - one property of exponential functions (\(y = a^x\) is one - to - one), \(x−5 = 0\), so \(x = 5\).

Step3: Analyze option B

Set \(f(x)=5^{x - 5}-5 = 0\). Then \(5^{x - 5}=5^1\). By the one - to - one property of exponential functions, \(x−5 = 1\), so \(x = 6\).

Step4: Analyze option C

Set \(f(x)=-5^{x - 5}+5 = 0\). Then \(5^{x - 5}=5^1\). By the one - to - one property of exponential functions, \(x−5 = 1\), so \(x = 6\).

Step5: Analyze option D

Set \(f(x)=-5^{x - 5}-1 = 0\). Then \(-5^{x - 5}=1\), or \(5^{x - 5}=-1\). Since \(5^{x - 5}>0\) for all real \(x\) (because the base \(a = 5>0\) in the exponential function \(y = 5^{x - 5}\)), there is no real - valued \(x\) that satisfies the equation \(5^{x - 5}=-1\).

Answer:

D. \(f(x)=-5^{x - 5}-1\)