\( \left\{\begin{array}{l}f(x)=x^{2}+4x-16 \\g(x)=2^{x}\end{array}\right. \)

We wish to find the values of \(x\) for which the two functions \[ f(x)=x^2+4x-16 \] and \[ g(x)=2^x \] are equal. That is, we must solve \[ x^2+4x-16=2^x. \] Define \[ h(x)=x^2+4x-16-2^x. \] We are looking for the roots of \(h(x)=0\). ### Step 1. Check an obvious candidate Try \(x=4\): \[ \begin{aligned} f(4)&=4^2+4\cdot4-16=16+16-16=16,\\[1mm] g(4)&=2^4=16. \end{aligned} \] Since \(f(4)=g(4)\), we have a solution at \[ x=4. \] ### Step 2. Look for a negative solution Because \(2^x\) is always positive, the behavior of \(h(x)\) for negative \(x\) can allow another intersection. We start by evaluating \(h(x)\) at a couple of negative values. - For \(x=-10\): \[ \begin{aligned} f(-10)&=(-10)^2+4(-10)-16=100-40-16=44,\\[1mm] g(-10)&=2^{-10}\approx\frac{1}{1024}\approx0.001,\\[1mm] h(-10)&\approx 44-0.001=43.999>0. \end{aligned} \] - For \(x=-6\): \[ \begin{aligned} f(-6)&=(-6)^2+4(-6)-16=36-24-16=-4,\\[1mm] g(-6)&=2^{-6}=\frac{1}{64}\approx0.0156,\\[1mm] h(-6)&\approx -4-0.0156=-4.0156<0. \end{aligned} \] Since \(h(x)\) changes sign between \(x=-10\) and \(x=-6\) (from positive to negative) there must be a root in that interval. We can narrow it down further. - For \(x=-6.5\): \[ \begin{aligned} f(-6.5)&=(-6.5)^2+4(-6.5)-16=42.25-26-16=0.25,\\[1mm] g(-6.5)&=2^{-6.5}\approx\frac{1}{2^{6.5}}\approx\frac{1}{90.51}\approx0.011,\\[1mm] h(-6.5)&\approx0.25-0.011=0.239>0. \end{aligned} \] Now we have: \[ h(-6.5)\approx0.239>0\quad \text{and}\quad h(-6)\approx-4.0156<0. \] Thus there is a root between \(-6.5\) and \(-6\). ### Step 3. Refine the negative solution Using linear interpolation between \(x=-6.5\) and \(x=-6\): Let \[ \Delta x=0.5,\quad \Delta h=0.239-(-4.0156)=4.2546. \] The fraction from \(x=-6.5\) to the zero is approximately \[ t\approx\frac{0.239}{4.2546}\approx0.056. \] Thus the approximate root is \[ x\approx -6.5+0.056\times0.5\approx -6.5+0.028\approx -6.472. \] ### Step 4. Conclusion The solutions of \[ x^2+4x-16=2^x \] are approximately \[ x\approx-6.47\quad \text{and}\quad x=4. \] Thus, the points of intersection of the functions \(f(x)\) and \(g(x)\) occur when \[ x\approx -6.47\quad \text{and}\quad x=4. \]