1. **Problem:** Determine if the function $f(x) = 3$ is even, odd, or neither. 2. **Recall definitions:** - A function $f$ is **even** if $f(-x) = f(x)$ for all $x$. - A function $f$ is **odd** if $f(-x) = -f(x)$ for all $x$. 3. **Check $f(x) = 3$:** - $f(-x) = 3$ - Since $f(-x) = f(x)$, the function is **even**. --- 1. **Problem:** Determine if $f(x) = x^{-5}$ is even, odd, or neither. 2. **Check:** - $f(-x) = (-x)^{-5} = -x^{-5} = -f(x)$ - Since $f(-x) = -f(x)$, the function is **odd**. --- 1. **Problem:** Determine if $f(x) = x^2 + 1$ is even, odd, or neither. 2. **Check:** - $f(-x) = (-x)^2 + 1 = x^2 + 1 = f(x)$ - So, $f$ is **even**. --- 1. **Problem:** Determine if $f(x) = x^2 + x$ is even, odd, or neither. 2. **Check:** - $f(-x) = (-x)^2 + (-x) = x^2 - x$ - $f(-x) eq f(x)$ and $f(-x) eq -f(x)$ - So, $f$ is **neither**. --- 1. **Problem:** Determine if $g(x) = x^3 + x$ is even, odd, or neither. 2. **Check:** - $g(-x) = (-x)^3 + (-x) = -x^3 - x = - (x^3 + x) = -g(x)$ - So, $g$ is **odd**. --- 1. **Problem:** Determine if $g(x) = x^4 + 3x^2 - 1$ is even, odd, or neither. 2. **Check:** - $g(-x) = (-x)^4 + 3(-x)^2 - 1 = x^4 + 3x^2 - 1 = g(x)$ - So, $g$ is **even**. --- 1. **Problem:** Determine if $g(x) = \frac{1}{x^2 - 1}$ is even, odd, or neither. 2. **Check:** - $g(-x) = \frac{1}{(-x)^2 - 1} = \frac{1}{x^2 - 1} = g(x)$ - So, $g$ is **even**. --- 1. **Problem:** Determine if $g(x) = \frac{x}{x^2 - 1}$ is even, odd, or neither. 2. **Check:** - $g(-x) = \frac{-x}{x^2 - 1} = - \frac{x}{x^2 - 1} = -g(x)$ - So, $g$ is **odd**. --- 1. **Problem:** Determine if $h(t) = \frac{1}{t - 1}$ is even, odd, or neither. 2. **Check:** - $h(-t) = \frac{1}{-t - 1} = \frac{1}{-(t + 1)} = -\frac{1}{t + 1}$ - $h(-t) \neq h(t)$ and $h(-t) \neq -h(t)$ - So, $h$ is **neither**. --- 1. **Problem:** Determine if $h(t) = |3^t|$ is even, odd, or neither. 2. **Check:** - Since $3^t > 0$ for all $t$, $|3^t| = 3^t$. - $h(-t) = 3^{-t} = \frac{1}{3^t} \neq h(t)$ and $\neq -h(t)$ - So, $h$ is **neither**. --- 1. **Problem:** Determine if $h(t) = 2t + 1$ is even, odd, or neither. 2. **Check:** - $h(-t) = 2(-t) + 1 = -2t + 1$ - $h(-t) \neq h(t)$ and $h(-t) \neq -h(t)$ - So, $h$ is **neither**. --- 1. **Problem:** Determine if $h(t) = 2|t| + 1$ is even, odd, or neither. 2. **Check:** - $h(-t) = 2|-t| + 1 = 2|t| + 1 = h(t)$ - So, $h$ is **even**. --- **Summary:** - 47: even - 48: odd - 49: even - 50: neither - 51: odd - 52: even - 53: even - 54: odd - 55: neither - 56: neither - 57: neither - 58: even