1. **State the problem:** Find the domain, range, asymptote, and y-intercept of the functions: $$f(x) = 5^x$$ $$g(x) = 5^x - 8$$ $$h(x) = 5^{x+4}$$ 2. **Recall properties of exponential functions:** - The domain of any exponential function $a^x$ where $a>0$ is all real numbers: $$\text{Domain} = (-\infty, \infty)$$ - The range of $a^x$ is $(0, \infty)$ because $a^x$ is always positive. - The horizontal asymptote of $a^x$ is $y=0$. - The y-intercept is found by evaluating the function at $x=0$. 3. **Analyze each function:** **For** $f(x) = 5^x$: - Domain: all real numbers $$(-\infty, \infty)$$ - Range: $(0, \infty)$ - Asymptote: $y=0$ - Y-intercept: $$f(0) = 5^0 = 1$$ **For** $g(x) = 5^x - 8$: - Domain: all real numbers $$(-\infty, \infty)$$ - Range: shift the range of $5^x$ down by 8 units: $$ (0, \infty) - 8 = (-8, \infty) $$ - Asymptote: shift the asymptote down by 8 units: $$ y = 0 - 8 = -8 $$ - Y-intercept: $$ g(0) = 5^0 - 8 = 1 - 8 = -7 $$ **For** $h(x) = 5^{x+4}$: - Domain: all real numbers $$(-\infty, \infty)$$ - Range: same as $5^x$ because horizontal shifts do not affect range: $$ (0, \infty) $$ - Asymptote: same as $5^x$: $$ y=0 $$ - Y-intercept: $$ h(0) = 5^{0+4} = 5^4 = 625 $$ 4. **Summary table:** | Function | Domain | Range | Asymptote | Y-intercept | |---|---|---|---|---| | $f(x) = 5^x$ | $(-\infty, \infty)$ | $(0, \infty)$ | $y=0$ | $1$ | | $g(x) = 5^x - 8$ | $(-\infty, \infty)$ | $(-8, \infty)$ | $y=-8$ | $-7$ | | $h(x) = 5^{x+4}$ | $(-\infty, \infty)$ | $(0, \infty)$ | $y=0$ | $625$ | This completes the analysis of the domain, range, asymptote, and y-intercept for the given functions.