1. **Determine values of a and b for** $f(x)=a\cos(x+b)$ from the graph. - The amplitude $a$ is the maximum value of $f(x)$. - From the graph, the maximum value is $2$ so $a=2$. - The cosine function normally has a maximum at $x=0$ for $\cos x$, but here the maximum is at $x=0$, so $b=0$. 2. **Graph of** $g(x) = \sin 2x + 1$ **on** $[-180^\circ, 180^\circ]$: - The function $\sin 2x$ has period $\frac{360^\circ}{2} = 180^\circ$. - The vertical shift is $+1$, so midline is at $y=1$. - Intercepts: set $g(x)=0 \Rightarrow \sin 2x = -1$, so $2x = -90^\circ, 270^\circ, ...$ leading to intercepts. - Turning points occur where derivative $g'(x) = 2\cos 2x = 0$, so $2x=90^\circ, 270^\circ, ...$ ; thus, turning points at $x=45^\circ, 135^\circ, ...$. 3. **Period of** $g$: - The period is $\frac{360^\circ}{2} = 180^\circ$. 4. **Range of** $2g(x)$: - Since $g(x) = \sin 2x +1$ ranges from $1 -1 = 0$ to $1 +1=2$, - Therefore, $2g(x)$ ranges from $2\times 0=0$ to $2\times 2=4$. - So the range of $2g(x)$ is $[0, 4]$. 5.1 **Values of** $x$ **for which** $f(x) < g(x)$ **in** $[-180^\circ, 0^\circ]$: - Using $f(x) = 2\cos x$ and $g(x) = \sin 2x + 1$, - Evaluating/sign comparing in the interval shows $f(x) < g(x)$ roughly between $-180^\circ$ and about $-120^\circ$, and near $-45^\circ$ to $0^\circ$. - Exact values depend on solving $2\cos x < \sin 2x + 1$. 5.2 **Values of** $x$ **where** $\tan(x + b)$ **is undefined for** $x \in [-180^\circ, 180^\circ]$: - $b=0$, so where $\tan x$ is undefined. - Tangent undefined where $\cos x = 0$, i.e., at $x = \pm 90^\circ$. 6. **Equation of** $p$ **shifted left by** $45^\circ$ from $g(x)$: - Original: $g(x) = \sin 2x + 1$. - Shift left 45°: $p(x) = g(x+45^\circ) = \sin 2(x+45^\circ) + 1 = \sin (2x + 90^\circ) + 1 = \cos 2x + 1$ (since $\sin(\theta + 90^\circ) = \cos \theta$). **Final answers:** - $a=2$, $b=0$ - Period of $g$ is $180^\circ$ - Range of $2g(x)$ is $[0,4]$ - $f(x)