What are the features of the function f, of, x, equals, left bracket, one half, right bracket, to the power x , minus, 6f(x)=( 21​ ) x −6 graphed below?

To analyze the function $f(x) = \left(\frac{1}{2}\right)^x - 6$ and understand its features based on the given equation $f(x) = (21)x - 6$, we will identify and break down the necessary features.

1. Break Down the Problem

We will:

1. Identify key features of the function $f(x) = \left(\frac{1}{2}\right)^x - 6$.
2. Investigate the intersection or relationship with the line $y = 21x - 6$.

2. Relevant Concepts

• The function $f(x) = \left(\frac{1}{2}\right)^x$ is an exponential decay function, since the base $\frac{1}{2} < 1$.
• The linear function $y = 21x - 6$ has a slope of 21, which is steep.
• We need to consider the intercepts, asymptotes, and behavior as $x$ approaches extremes.

3. Analysis and Detail

1. Function $f(x) = \left(\frac{1}{2}\right)^x - 6$:

   • Y-Intercept: Set $x = 0$ $f(0) = \left(\frac{1}{2}\right)^0 - 6 = 1 - 6 = -5$
   • X-Intercept: Solve $f(x) = 0$ $\left(\frac{1}{2}\right)^x - 6 = 0 \Rightarrow \left(\frac{1}{2}\right)^x = 6$ Taking logarithms gives: $x \log\left(\frac{1}{2}\right) = \log(6) \Rightarrow x = \frac{\log(6)}{\log\left(\frac{1}{2}\right)} \approx -2.585$
   • Asymptote: The horizontal asymptote is $y = -6$ as $x \to \infty$.

2. Linear Function $y = 21x - 6$:

   • Y-Intercept: When $x = 0$, $y = -6$.
   • X-Intercept: Set $y = 0$: $0 = 21x - 6 \Rightarrow x = \frac{6}{21} = \frac{2}{7} \approx 0.286$

4. Verify and Summarize

• The exponential function decreases from $-5$ at $x = 0$ and approaches $-6$ but never reaches it.
• The linear function passes through $-6$ on the y-axis and increases without bound.

Final Answer

The features of the function $f(x) = \left(\frac{1}{2}\right)^x - 6$ include its decay behavior with a y-intercept at $-5$, a horizontal asymptote at $y = -6$, and an x-intercept approximately at $-2.585$. The linear function $y = 21x - 6$ crosses the y-axis at $-6$ and has a steep slope of $21$. The two functions intersect at points which would need further numerical methods to calculate explicitly.

If needed to solve or find intersection points further, one would set $\left(\frac{1}{2}\right)^x - 6 = 21x - 6$ and solve for $x$.