Question

1. $f(x) = 5^x$

x	f(x)
-2	0.04
-1	0.2
0	1
1	5
2	25

(grid graph)
domain: $-infty, infty$
range: $0, infty$
asymptote: 0
y-intercept:
growth or decay? growth

Explanation:

Step1: Find the domain

The domain of an exponential function \(y = a^{x}\) (\(a>0,a
eq1\)) is all real numbers. For \(y = 5^{x}\), we can substitute any real - valued \(x\) into the function. So, the domain is \((-\infty,\infty)\).

Step2: Find the range

Since \(5^{x}>0\) for all real \(x\) (because any positive number \(a = 5\) raised to a real power \(x\) is positive). As \(x\to-\infty\), \(y = 5^{x}\to0\) (but \(y
eq0\)), and as \(x\to\infty\), \(y = 5^{x}\to\infty\). So the range is \((0,\infty)\).

Step3: Find the asymptote

The horizontal asymptote of the exponential function \(y=a^{x}\) (\(a > 0,a
eq1\)) is \(y = 0\). As \(x\to-\infty\), \(y=5^{x}\to0\).

Step4: Find the y - intercept

The y - intercept occurs when \(x = 0\). Substitute \(x = 0\) into \(y=5^{x}\), we get \(y=5^{0}=1\) (using the rule \(a^{0}=1,a
eq0\)).

Step5: Determine growth or decay

For an exponential function \(y=a^{x}\), if \(a>1\) (here \(a = 5>1\)), the function is an exponential growth function. As \(x\) increases by \(1\), the value of \(y\) is multiplied by \(a\). For example, when \(x = 1,y = 5\); when \(x=2,y = 25\) (\(25\div5 = 5\)).

Answer:

Domain: \((-\infty,\infty)\)
Range: \((0,\infty)\)
Asymptote: \(y = 0\)
y - intercept: \(1\)
Growth or Decay: Growth