Q. Which is the graph of \(f(x)=100(0.7)^x\)?

Q: How long until the value halves (half-life)?

Solve 100(0.7)^{x} = 50 ⇒ (0.7)^{x} = 0.5 ⇒ x = ln(0.5) / ln(0.7) ≈ 2.81 units.

Answer

Explanation: The function \(f(x)=100(0.7)^x\) is an exponential decay.

Evaluate a few points:

\[ f(0)=100, \quad f(1)=100(0.7)=70, \quad f(2)=100(0.7)^2=49 \]

Hence the curve is decreasing, always positive, and has a horizontal asymptote at \(y=0\).

Domain: \((-\infty, \infty)\)
Range: \((0, \infty)\)

Final result: A decreasing exponential curve starting at \((0,100)\), passing through \((1,70)\) and \((2,49)\), and approaching \(y=0\) as \(x \to \infty\).

Detailed Explanation

Problem: Analyze the function \(f(x)=100(0.7)^{x}\)

Step 1 – Identify the type of function

The function is of the form \(f(x)=ab^{x}\) with \(a=100\) and \(b=0.7\). Since \(0Step 2 – Find the y-intercept

Evaluate at \(x=0\):

\[
f(0)=100(0.7)^{0}=100\cdot 1=100
\]

The graph passes through the point \((0,100)\).

Step 3 – Behavior as \(x\) increases

As \(x\) increases, \((0.7)^{x}\) decreases toward 0, so

\[
\lim_{x\to\infty}f(x)=\lim_{x\to\infty}100(0.7)^{x}=0.
\]

For example, \(f(1)=100(0.7)=70\). The x-axis \(y=0\) is a horizontal asymptote.

Step 4 – Behavior as \(x\) decreases

As \(x\) becomes negative, \((0.7)^{x}\) grows without bound, so

\[
\lim_{x\to -\infty}f(x)=\infty.
\]

For example, \(f(-1)=100(0.7)^{-1}=\dfrac{100}{0.7}\approx 142.857\).

Conclusion

The graph is an exponential decay curve that is very large on the left, passes through \((0,100)\), and decreases to the right toward the horizontal asymptote \(y=0\) without touching it.

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FAQs

Is f(x) = 100(0.7)^{x} exponential growth or decay?

Decay, because the base 0.7 is between 0 and 1. Values decrease as x increases.

What is the domain and range of f(x) = 100(0.7)^{x}?

Domain: all real numbers, (-∞, ∞). Range: positive reals, (0, ∞), since exponential never reaches 0.

What is the y-intercept?

f(0) = 100(0.7)^{0} = 100, so the graph crosses the y-axis at (0, 100).

Does f(x) have an x-intercept?

No. 100(0.7)^{x} is always positive, so it never equals 0 and has no x-intercept.

What is the horizontal asymptote?

The horizontal asymptote is y = 0; as x → ∞, f(x) → 0.

What is the end behavior?

As x → ∞, f(x) → 0. As x → −∞, f(x) → ∞ (it grows without bound going left).

How do I graph it quickly?

Plot the y-intercept (0,100), a few points: (1,70), (2,49), (−1,100/0.7 ≈143), draw a smooth decreasing curve, and show horizontal asymptote y = 0.

What is the percent decay per unit?

The decay factor is 0.7, so the percent decrease per unit is (1 − 0.7)×100% = 30% per unit.

How do I solve 100(0.7)^{x} = y for x?

Use logarithms: x = ln(y/100) / ln(0.7). Note ln(0.7) < 0, so signs invert accordingly.

How long until the value halves (half-life)?