QUESTION 5 KZNJ16 The equation $$ f(x)=a^{x}, a0, a
eq 1 $$, passes through the point $$ \left(-3 ; \frac{1}{8} ight) $$. $$ 5.1 \quad $$ Determine the value of $$ a $$. $$ 5.2 \quad $$ Write down the equation of $$ f^{-1} $$ in the form $$ y=\ldots $$ $$ 5.3 \quad $$ Determine the value(s) of $$ x $$ for which $$ f^{-1}(x)-3 $$.

Question

QUESTION 5 KZNJ16 The equation $$ f(x)=a^{x}, a>0, a 
eq 1 $$, passes through the point $$ \left(-3 ; \frac{1}{8}ight) $$. $$ 5.1 \quad $$ Determine the value of $$ a $$. $$ 5.2 \quad $$ Write down the equation of $$ f^{-1} $$ in the form $$ y=\ldots $$ $$ 5.3 \quad $$ Determine the value(s) of $$ x $$ for which $$ f^{-1}(x)>-3 $$.

UpStudy AI · Accepted Answer

Let's address each part of the question step-by-step.

---

### **5.1. Determine the value of $$ a $$.**

Given the exponential function:
$$
f(x) = a^{x}, \quad a > 0, \quad a \neq 1
$$
and it passes through the point $$ \left(-3, \frac{1}{8}\right) $$.

**Substitute the point into the function:**
$$
f(-3) = a^{-3} = \frac{1}{8}
$$

**Solve for $$ a $$:**
$$
a^{-3} = \frac{1}{8} \quad \Rightarrow \quad a^{3} = 8 \quad \Rightarrow \quad a = \sqrt[3]{8} = 2
$$

**Answer:**
$$
a = 2
$$

---

### **5.2. Write down the equation of $$ f^{-1} $$ in the form $$ y = \ldots $$**

The function $$ f(x) = 2^{x} $$ is exponential. To find its inverse, follow these steps:

1. **Start with the original function:**
$$
y = 2^{x}
$$

2. **Swap $$ x $$ and $$ y $$:**
$$
x = 2^{y}
$$

3. **Solve for $$ y $$:**
$$
\log_{2} x = y
$$

**Thus, the inverse function is:**
$$
f^{-1}(x) = \log_{2} x
$$

**Answer:**
$$
y = \log_{2} x
$$

---

### **5.3. Determine the value(s) of $$ x $$ for which $$ f^{-1}(x) > -3 $$.**

We have:
$$
f^{-1}(x) = \log_{2} x > -3
$$

**Solve the inequality:**
$$
\log_{2} x > -3
$$

**Exponentiate both sides with base 2:**
$$
x > 2^{-3}
$$
$$
x > \frac{1}{8}
$$

**Domain Consideration:**
Since $$ \log_{2} x $$ is defined only for $$ x > 0 $$, the solution must also satisfy this condition.

**Answer:**
$$
x > \frac{1}{8}
$$

---

### **Summary of Answers:**
1. **5.1:** $$ a = 2 $$
2. **5.2:** $$ y = \log_{2} x $$
3. **5.3:** $$ x > \frac{1}{8} $$
**5.1.** $$ a = 2 $$ **5.2.** $$ y = \log_{2} x $$ **5.3.** $$ x > \frac{1}{8} $$

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